First I should mention that I change my strings every 6 months on guitars that I rarely play and every month or two on the ones that I play often. Even strings that are not played will lose tone due to oxidation of the metal strings.
If your high e string stays in tune when played open, but plays out of tune when fretted, then you most likely have something going on that is affecting your guitars intonation. Proper intonation is what makes a guitar's strings play in tune as you fret different notes on the fret board. Intonation revolves around changes in the strings length when it is pressed against a fret. The amount of change in the string length determines the amount of change in the pitch when the string is played.
To check the intonation, use a digital tuner and play the open string and be sure it is in tune. Then play the same string on the 12th fret (same note one octave higher). If the 12th fret note is flat or sharp, your intonation is off.
On most electric guitars you can adjust the intonation for each string individually at the bridge. However, an acoustic guitar does not offer an intonation adjustment device for individual strings. Acoustic guitars ideally are adjusted for intonation at the factory and most acoustic guitars have a compensated saddle that is also tilted to account for the need for the thinner strings to be slightly shorter in overall scale than the fat strings in order to maintain intonation. Unfortunately, it's difficult to achieve perfect intonation on an acoustic, but it sounds like your guitar has some significant problems.
There are several possible things that might be contributing to your intonation issue. And some would also explain the fret buzz. Here are some things to check:
A significant change in string gauge can affect intonation to some extent as well as contribute to fret buzz. When the guitar was originally set up in the factory, the intonation was set as best as the design of the guitar would allow. Saddle compensation, neck relief and the action (string height above fretboard) have been set for a particular gauge string. A string with a different amount of tension can contribute to intonation problems.
Too much or too little neck relief will change your intonation. Neck relief is the amount of bow in the neck and can be adjusted by the truss rod. Changing string gauge will usually necessitate a truss rod adjustment due to the change in overall string tension exerted on the neck. Going to heavier strings will exert more pressure and without an adjustment to the truss rod to compensate, will create more neck relief. In other words the neck will be bowed more. Going to lighter strings will allow the neck to straighten more due to decreased tension. If the neck is too flat or has a back bow, you will get fret buzzing as well as poor intonation. Intonation is affected by the amount of change in the string length when your fret a note. The neck relief affects how far you have to move the string to reach the fret and thus is related to intonation.
A loose or leaning saddle can change the length of the string and affect intonation. Be sure that the saddle is properly positioned in the slot in the bridge and not leaning either way. If you take all the strings off at once to re-string, it is possible for the saddle to change position.
Grooves in the saddle can also change the length of the string and affect intonation. Some saddles are plastic and easily develop grooves, particularly under the plain steel b and e strings.
Grooves in the frets can also change the length of the string and affect intonation. They can also cause fret buzz. The more you play, the more likely you are to begin wearing grooves in some of your frets. This is particularly problematic on the frets under the plain steel b and e strings because the friction of steel against the softer nickel frets eventually wears grooves in the frets. Another thing that happens is some frets can become worn and lose their crown. Frets are crowned to provide a small isolated contact point to aid with proper intonation. If the frets become worn and flat, your string length will change because the contact point on the fret has changed and your intonation will be off. If you have some deep grooves, or some of your frets are worn down, you may need to have the frets dressed, and re-crowned. Eventually if you play your guitar often enough, you will wear the frets down to the point where you need to have them all replaced (known as a re-fret).
These are a few things to check and there may be more than one of the above issues that can be contributing to your particular problem. One other thing you might consider is to simply change the strings again and see if that helps. It's possible that you could have a defective string.
The best advice I can give you is to not only get another set of good quality new strings, but to take your guitar to a luthier or qualified guitar repair technician for a proper set up based on the string gauge you choose. During the set up, most of the issues discussed above will be addressed or discovered. And if there is something more serious going on (such as improper saddle compensation), they will be able to tell you what needs to be done.