Acoustics tend to have fatter necks than electrics, that's a huge problem for me, after a tendon was severed when I was 12 and surgically reconnected. I have a lot of trouble finding an acoustic that doesn't hurt my entire forearm, but mostly the wrist.
I always set the bridge height (not the truss rod, never touch it unless it absolutely needs to be adjusted for proper backbow) so that the strings are as close to the frets as I can get them without buzzing. Some bolt on necks are made so they tilt, the action is adjusted there instead of lowering the bridge, but that's usually on electrics, not acoustics. Usually you have to file the bottom of the saddle down to lower acoustic action. (that's the proper name for the white bone or plastic piece the strings sit on at the bridge) The truss rod has to be set for a slight backbow, so that it bows away from the frets in the middle, and if your hand is giving you trouble switch to a lighter gauge strings. If it's still a problem, find a guitar with a slimmer neck. That's the main thing for me, I have to search for slim necks, and I never even think about purchasing an acoustic until I play it several times, 30 minutes each time and be sure it won't cause pain once I get onstage with it.
You can check it pretty easily, sight down the neck from the tuning key end, if you see the neck very slightly bow away fromt he strings in the middle, the truss rod is OK. Capo at the 1st fret and fret at the last, it should have around .010-.012" clearance in the middle, .015" at most, you can check it with a feeler gauge. If you see an upbow, toward the strings, that needs to be adjusted, of it's straight and does not buzz, leave it alone. A straight neck is not a problem, but will tend to buzz if you drop the strings very close to the frets. don't touch the truss rod unless it is definitely needed, it's really easy to break and the repair is more expensive than most guitars, unless you happen to have one that's worth more than $500 or so...and are really attached to it...it's not worth it. Most local guitar techs can't even think about doing it either, you'll have to send it off, probably to the factory.
Strings open and unfretted, tuned to concert pitch, you should have no more than about 1/8 inch clearance at the octave, maybe a 64th more or so. I like a bit less for an acoustic. Around 3/32-7/64" is good for me. If it's higher, pull the saddle and sand the bottom lengthwise LIGHTLY. Replace and check, repeat until it gets good action but does not buzz. If you sand too much, it can be shimmed with a thin plastic strip, or brass if you can get some shim stock from a machine shop. I was a machinist for 8 years, I still have some .001" - .015" shim stock lying around that I use.
My guess is you have a guitar with a fat neck and the strings are too high.
Right now I have a Takamine that is great, and most Ovations I've played have slim enough necks I can pick up just about any one and play it. Most Takamines too have usable necks for me, most other brands I have to be really picky. Haven't played any Taylors though, only one. it seemed to have a nice neck but I only played it for a couple of minutes, not long anough to really tell.
Get your guitar set up better, lower the strings, switch to a lighter gauge, and some of the suggestions already made are worthwhile too. My thumb is all over the place, depending on what I do. Sometimes I wrap it around the neck, sometimes I use it to assist with barre chords, sometimes it's barely touching at all.
I'm a member of the BOMB SQUAD.
If you see me running, better catch up!
http://billy-griffis-jr.artistwebsites.com/