Yep. Stew Smith's "Push Up Push" program worked alright for me:
Basically 200 push ups/day alternating days between lots of easy sets (about 1/2 of your current max reps per set) and trying to get through it in as few sets as possible (I'd still stop well short of failure). Boring as hell, but pretty effective I found. Push ups are also a skill. Scapular retraction and depression, total body tension (especially core and posterior chain activation), breathing for increased intra-abdominal pressure, trying to corkscrew your hands outward (like breaking a branch over your leg) so you can "push from your armpits" engaging your lats. These are all cues that helped me. I consciously practiced them as opposed to just grinding the reps out thoughtlessly.
I would add an equal number of band pull aparts, some broom handle dislocates and lots of YTWL's to keep your shoulders happy. I would sub crunches for sit-ups to spare your back, but I see no reason the same principle wouldn't work.
I'd take a break from much else in the way of upper body pressing movements unless your shoulder health/recovery are excellent.
Get lean if you aren't already.