I'm nonbinary, but people seem to be weirded out that I like my tits and want to keep them. I wear a chest binder sometimes with some of my clothes because they fit better that way, but it's a lot of work to acclimate to wearing a binder full-time and I don't see the point when I don't have a problem with them.
For the record, I don't think I'm necessarily that big (I've definitely seen bigger on people otherwise my size) but the bra industry decided I am because I have a narrow back. I literally learned to make my own because it's cheaper than buying bras and being disappointed in them. (And then I went into making clothes generally, and if you ever get into that it is an eye-opener to so. many. things.)
The shitty truth is, OP, if you go shopping for bras you'll struggle to find anything you like. If you're lucky and you have the money for it, find someone who can do custom bras or better yet, something like short stays (basically a waist-length corset) or vintage-style long-line bras (virtually the same thing but with a modern bra closure). It's expensive, but they'll last longer and help a lot with the pain. (Full corsets are another option, but they can be a bit limiting. There's one really good YouTube video about wearing a corset full-time for several years that does discuss the drawbacks very honestly.)
In store-bought bras, if that's all you can do, look for sports bras but make sure the band is big enough and it has wide straps. If you're getting any pain in your sternum or ribs, the band is too small.
Most commercial bras, even sports bras, for larger cup sizes (so like, everyone above a C apparently) are way too narrow in the straps, which concentrates all the pressure into narrow points on your shoulders, which is what leads to a lot of pain. A lot of modern long-line bras aren't long enough either and use shitty cheap reinforcing that can't actually support the weight of your chest properly, so it all goes on the straps (which are usually narrow).
As for exercise... sports bras are a big help, but they need to fit really, really well or they don't work, and I don't think you'd find anything that would work for running or aerobics or anything like that. Yoga isn't too bad, I didn't find weightlifting or Pilates to be too bad either. For cardio I stick to cycling because my knees hate running, and that's not too bad but it did take a while to find a bike that was comfortable because I prefer to sit more upright rather than hunching over.
Yoga is especially good because there's some good core and shoulder work you can do that'll help with tension and strengthening. Getting regular deep-tissue massages in your neck, shoulders and upper back also goes a long way for pain issues.
BUT. With that being said, if they're always hurting (not sensitive, actually painful) then definitely go see a doctor OP. It's not usually a good sign, so best to rule out any actual problems.