This sounds similar to my situation. I have been doing 5/3/1 for a couple of years. Pulled my back once (popped during a deadlift) and it made me a bit gunshy around that exercise. Gave the back a rest then started focusing on my core and glutes. It has helped my form a lot and given me more confidence in my deadlifts.

Also had the right shoulder impingement (rotator cuff) and couldn't sleep on my right side. Thought it would heal on it's own, but after a couple of months, I finally bit the bullet and went to a physical therapist. Was surprised by their diagnosis and where they wanted me to foam roll was back near the scapula and upper back, not where the pain was on the outer shoulder. Did the exercises and stretches they told me to and it cleared up in about a month. I can also tell when that area is getting tight now and I use their exercises for preemptive maintenance. Haven't had a flair up in about a year and have my overhead press back up to 120. I recommend the PT and doing exactly what they prescribe if you haven't given that a shot yet.

As far as getting bored of beginner 5/3/1. I recommend grabbing the book - 5/3/1: The Simplest and Most Effective Training System for Raw Strength (2nd Edition). It has a lot of variations on the 5/3/1 routine and talks about how 5/3/1 for beginners is really an intro to the routine that initially focuses on a more full body workout. I still do the 4 main lifts on the regular, but there is a healthy amount of variation in the accessory work to keep it interesting.

You can always re-define your goals if you are bored. Flexibility, body fat percentage, aesthetics, strength, particular muscle groups, endurance, or stamina. I personally really like having the data so I can see my progress and that really motivates me. I use Fitnotes for exercise tracking. Hope you can find something that works well for you.