One of the most important things you can do is find a teacher who has a track record of producing students who can sound like you want. You don't have to study with them weekly either. Sometimes it just takes someone who really knows how embouchures work to point out one or two things and you finally reach the peak of that initial plateau in sounding good. This is exactly what happened in my personal experience. I didn't make as much improvement as I would have liked throughout my undergraduate. It wasn't until I took a few lessons with a DC Freelancer (Doug Elliott) and switched to a slightly larger rimmed mouthpiece than I had ever tried, that I began to be able to be happy with the sound that I was producing. Fortunately, this is easier than ever with Skype. Many good players give internet lessons, including the guy I mentioned earlier.
Recording equipment may also play a role in how you sound. Lindberg recordings have a LOT of reverb. I don't know what his sound engineer was going for to be honest.He sounds great in person, but I've never really been a huge fan of the recordings because of that. However, some of the magic you hear is because of mic placement, messing with the EQ etc. Certain devices do a better job naturally of recording than others. I've found, for example, the Zoom H4 to be really good at just being ready to go to record after you change a few settings when you turn it on. The cheapo 1/8" aux mic that I plug directly into my computer with no EQ and record into audacity... not so much.
To actually directly answer the question: I've found that the quality you are describing is almost exclusively due to many overtones being produced in the sound. It is almost entirely just playing in tune. But I don't necessarily mean the same thing as playing with a tuner and having the green lights come on. I can play the right pitch amplitude with a really weak sound and get the tuner to light up. What I can't do is get that same really weak sound to blend with a drone CD.
I personally use the Tuning CD - you would want to maybe get a different one depending on country. EU and most other parts of the world would want the 442HZ version. US and others that use US tuning the 440 would be best. Although it's almost imperceptible the difference, and practicing with either won't hurt you.
Basically what this tape does is emit several octaves of the same pitch, which creates overtones. When you play a pitch over it, it will be obvious if you are both playing out of tune and if your overtones aren't lined up. Pitch can be very precise. After playing individual pitches on a drone tape, you'll likely know exactly what I'm talking about.
Further, when you play something other than pitches on it, say a Bb major scale over the Bb track, it makes it very easy to hear where the note wants to "go". IE: When you play the A natural in that scale, it sounds very dissonant, although you can still hear when the overtones aren't lined up with the Bb. When you play an arpeggio, you'll note that the 3rd and 5th (D and F, respectively in Bb major) will be in slightly different places than you would normally play them in. The 3rd needs to be played in a low 4th position, and the F needs to be played probably as close to the barrels as your trombone will allow.
When you play with the drone tape, I also will sometimes push my tuning slide all the way in. This means that to play in positions I would normally play in, I'd have to play a little bit lower. This gives me slide vibrato in 1st position (sort of important if you do jazz or commercial stuff as I do), but more importantly, it allows my embouchure to play the same pitches in a more relaxed fashion. Note: When you shorten an instrument, you also shorten the overtones that are created as well as the length of each position. Pay attention to this as you're paying with the drone. If you play melodies over the drone, get to the point where you can switch between notes without needing to take time to adjust after each note. After practicing like this for even a week, you will definitely begin to notice that your ability to blend with other musicians will be much better, and you'll likely be closer to the sound you want to hear.
Also! You can ask both of the players that you mentioned themselves for advice too!
Joe Alessi has a contact page:
(Scroll down to the part that says Write a Comment or Question
Another useful resource is the tromboneforum, which has several threads that you may find helpful:
Whats the proper way to record trombone?
Tone quality- so frustrated
EDIT: To respond to the recording comment:
Hate to disappoint you more, but everybody you hear is "not 100% themselves" unless you are hearing in person! That's not necessarily a bad thing. it's just a reality of recording. Mic placement determines how much of the "room" you hear on a recording. Guys like Watous 'eat the mic', meaning the mic goes INTO the bell when they record. 0% room ambience! You can still hear their tone quality, though. But the EQ settings can change the highs and lows and produce much differing sounds. Take two tracks I recorded for example:
^This was recorded in my room at the time, which had very little reverb and I did no adjustment to the EQ as well.
^This was done in the same room a few days later, but I spent more time adjusting the EQ and ended up posting it awhile later. This one changes the reverb settings and some other things to highs/lows.
Neither were "100% me" technically speaking, although they are pretty close depending on what room I'm playing in. Without adjusting for that though, it is not nearly as pleasant to listen to! Even with the adjustments, I'd much rather listen to Lindberg doing what I did. For further comparison, check out Jim Nova's overdubs:
There is a little additional assistance in that he has a really good mic, although I used a decent mic for the stuff I posted, but it's quite evident that his tone quality is distinct from mine even with me doing EQ changes!