You don't need to be training at 90%+ to get stronger, but at the same time the standard 5/3/1 program is mostly geared towards hypertrophy until you start to stall. What is your weight and height? Are you trying to gain mass? Personally, I don't do more than 5 reps (at least 1 or 2 away from failure) on the competition lifts and save the high rep stuff for variations and accessory work.
If you're in constant pain anytime you are squatting or pressing then you could consider switching to SSB squats for a few weeks, if you have access to a SSB. Part of the problem is inflammation, if you don't let the inflammation subside then at best it just wont get any worse. Like I said, for a lot of people there is no way to solve this problem and the only solution is to save low bar squatting for peaking phases and meets. Eric Cressey says that it has do to with the shape of the scapula, some people are predisposed to this kind of thing. Blaine Sumner said that he only squats up to a top single these days, his shoulders and arms can't take reps anymore and he does a lot of belt squats and other stuff.
Things that may help are pre-workout ibuprofen, doing barbell curls (you want your hands supinated) a few times a week, and lots of external rotation work - band pull aparts and face pulls are particularly good because they don't cause much fatigue and it's easy to get a good amount of volume in. If you simply increase rows and chin-ups (which you probably do lots of since you are on 5/3/1) then you will cause a lot more fatigue that can affect the lifts that actually matter.