OP has Tokina 12-24, which is fine. Yes, 11-16 is a faster f2.8 lens, but narrower range of focal lengths, and a higher price too. Shooting mostly static subjects, especially on a tripod and stopping down for max depth of field anyway, f2.8 is wasted.
And, OP's 18MP T2i would be fine in many situations. It's the same sensor and image quality as T3i, 60D and 7D, so you'd see no improvement at all from those.
For really large prints and max possible quality, among Canon cameras the 5DII would be better. It would give you more fine detail in those really large prints. You'd have to step up to medium format such as Hasselblad or Leaf/Mamiya to do much better, currently.
A used 5D classic is considerably cheaper and a good camera, but isn't going to give you the resolution you're getting with the T2i now. Hard to say if large prints would improve much or at all. I'd try to get 5DII, if at all possible.
Yes, lenses are the other big factor. OP's lenses aren't bad, by any means. But there might be room for improvement a little at a time. This might be an ideal use for tilt shift lenses, for example.
If you switch to full frame 5DII, the only lens you have that's technically incompatible is the Tokina 12-24. I can tell you it's usable on the 5DII, though in a limited way. It won't start vignetting until about 18 or 19mm wide. However, you'll likely want to replace it with a wider, full frame lens eventually... such as 17-40 or 16-35. Actually, I use 20/2.8 and am happy with that.
You might explore some other techniques, such as stitching together some panoramas (vertical is possible, too), and/or or HDR. Flash might be an interesting effect in some cases, but I'd use it sparingly... it won't look natural in the woods. Do you have a circular polarizer? That can really help saturate the greens of foliage, especially on overcast day.