I have an ovation acoustic guitar. It's only about a year or two old. However it's always had terrible action (though this might be an issue with its been width). It was so bad that I was tuning to D standard and had fairly light strings (.12 gauge). I got it set up at a guitar center recently. They told me they could only lower the action a little bit. I believe they said the truss rod could only be lowered half a step? They told me that switching between an open D and D standard was messing up the tross rod and if I continued to switch between tunings, fixing the action would require much more money. Is this true? I also tend to do this on my electrics but the action isn't so bad on them.
Lots of misperception here!
Besides changes in humidity, using different gauge strings is the next-largest single contributor to action changes/problems, as is tuning the entire guitar up or down a whole step or more.
Using alternate open tunings will typically have very tiny, mostly-irrelevant affects on the playing action.
The purpose of the truss rod is to counteract the many pounds of pull of the strings on the neck. Tightening it bends the neck back (lower action) and loosening it bends the neck forward (higher action). The height of the bridge also matters a lot, and my guess is that this is your actual culprit.
Also...the only real reason for using multiple guitars for multiple tunings is to minimize the amount of time your audience spends listening to "the tuning song" during your set. It really busts up the flow of a performance. A modern, well-constructed guitar doesn't have any issue with changing tunings on-the-fly.
Also...if the tech saw that he had no more room to tighten the truss rod, and you're using light gauge strings, and the action is objectionably high, it's entirely possible that you have a broken truss rod.
If the neck is already straight, the action can be lowered by lowering the saddle: loosen the strings, remove the saddle from the bridge, and file the bottom of the saddle.
Filing should be done carefully and in a straight line, by laying sandpaper on a flat table and rubbing the bottom of the saddle on it, so that the bottom of the saddle remains straight.
Also, do it a little bit at a time and repeatedly test it until you get it the way you want it.
An even better option is to buy a few spare saddles, so you can test without having to worry about going too far. If you go too far with a test saddle, place something under it, until you figure out the perfect height, and then file your main saddle so that it achieves exactly that action.
A little confusion here! .012" set of strings isn't 'fairly light' - it's about standard. The truss rod doesn't get adjusted in 'steps'. If they meant half a turn it's still meaningless. The truss rod adjustment is only part of sorting out the action. The bridge height is just as relevant. As is the effect of heavy strings. They won't change the action, once they're on, but they will make it harder to play. Suggest taking it to another guitar tech., who may look at it differently. Changing tunings WILL affect the action, rather than the truss rod, so maybe, on that particular guitar anyway, doesn't come highly recommended.
Switching tunings can definitely affect your action - particularly if you are changing the string tension more than a semitone or half step on any given string. I know this is true because I have had numerous guitars set up for lowest action without buzzing. I like to tune my guitars a half step flat (D#). I tell my luthier that - but if he forgets and sets up my guitar for standard tuning, I get buzzing and I have to go back and have it adjusted again for the drop tuning I use.
Having said this, you can have your guitar set up with higher action, and it will play buzz free in many alternate tunings with no further adjustment between tunings. I know several guitarist who regularly use four or more different alternate tunings on the same guitar in a live show. They only change the tuning between songs - not the set up.
As far as what it can do to your truss rod - if every time you change tunings, you adjust your truss rod for that specific tuning, you could in fact wear out the adjustment mechanism by adjusting it back and forth too often. It's mechanical and can wear out if adjusted too often. But it is unlikely that you would be adjusting the truss rod between tunings.
If you do find that for your playing comfort and to keep the action as low as possible and not have any fret buzz, that you are feeling a need to adjust the truss rod between alternate tunings, I would definitely recommend a second guitar for the alternate tuning.
Again, many professional performing guitarist get away with one guitar and use that guitar for many alternate tunings during a show. Other professional guitarist prefer not to take the time to change the tuning - so they have two or more guitars on stage for their show - each tuned differently. In the extreme, there are a few artist that almost use a different guitar for every song in their show - but they are at a popularity level where they get their guitars for free and they have a person whose job is to bring them the right guitar for each song and they use a wireless instrument receiver.
If there is nothing wrong with your guitar, you should be able to find a compromise set up that will work for any of the tunings you use. If you want the lowest possible action for each tuning, I recommend using multiple guitars. I don't think it makes sense to adjust your truss rod every time you change tuning.
Switching tunings will not effect it as much as people think, but that's not to say it cannot or will not.
You do hear that if you tune your guitar to, oh let's say Open E, that it can hurt the guitar if left in that tuning. This may be true depending on the type of guitar, and if the guitar is left in Open E.
The more imminent issue will be string life. Constantly changing tunings will shorten the life of the strings. It's like anything metal. If you bend a paperclip back and forth it will eventually break. Changing tunings, stretches and relaxes the strings. Granted the effect isn't as quick as bending a paperclip back and forth until it breaks but each stretch weakens the tensile strength of the string.
That being said, I assume this is why some artists will pick one instrument, and leave it in the alternate tuning they use the most. An electric guitar can handle constant tuning changes better than an acoustic guitar. The strings will still have the same issue as acoustic guitars.
If you use lighter strings on the acoustic guitar, that will help reduce any possible guitar issues. Another thing you can do, instead of tuning to Open E, a lot of guitarists will tune to Open D, and then use a capo at the 2nd fret. This method puts less tension on the strings, and you get 2 different tunings you can use. I have one of my electric guitars tuned to Open E. I use lighter strings anyway so that combined with it being an electric guitar, seems to have eliminated any possible guitar issues. I haven't had an issue with it yet.
As far as action goes, the action can be affected by changing tuning: for example if you go to a tuning where, you are tightening, 3 of the 6 strings to get the desired notes for the new tuning, you are placing more tension on the neck, and it can increase the bow. This can be countered with a truss rod adjustment in most cases.
Which is another reason some guitarist will chose one guitar, and leave it in the desired tuning. If your a gigging musician, you don't have time to make truss rod adjustments between songs.
To answer the question in the title: Yes, somewhat. The truss rod wants to pull back. The strings want to pull forward. But the changes should be slight.
But the answer in the question text: Different styles of playing require different actions, and with loud strumming of cowboy chords, you want a higher action so the strings don't rattle against the frets. It can go too far, and it sounds like your guitar has gone that way.
If you can capo at the first fret and fret the last, there should be some room at say the tenth fret between the fret and the string. This is called relief and it should be small. IIRC about a dime's worth, but look online for guitar setup guides for more info, and turn the truss rod a quarter-turn at a time.
If that's about right, your guitar neck is fairly straight and the problem is your bridge. You could try to sand down the bottom of the saddle to bring it closer, or find a better repair shop than the one you used.
You say it's a two-year-old Ovation, and while even great guitars can need some adjustment to get it right for you, I'm frankly surprised that a new guitar from a reputable brand is so unmanageable. Good luck.