Changing rep ranges every couple weeks?

That's one way to do it, but I prefer varying the loads across the week.

It really depends, how long were you plateued for?
A month plateu, or even 2month plateu isn't that long. Theres only so much you can mix it up.

Also 3x5/5x5 isn't that much of a mix up, in my opinion, a better option would be 5x3 / 3x3 or something similiar.
With 3x5/5x5 it's the same rep range being used, your just changing the sets.

How long have you been training? If it's over a year, in my experience of talking to people, they seem to be okay  with more or less the same stuff for a year, and after that, may need a complete rehaul, provided the first year was a good progressive overload based routine.

Summary;
Try triples instead.
However, be sure to warm up, alot! Get some blood in those shoulders, dislocations/band pull aparts etc.
For me, I find triples alot easier to tweak something in my shoulders if they aren't properly warm, where as 5's have little consequence.

Certainly mixing up rep ranges is beneficial for hypertrophy, for example the HST program is based around that. Not sure for pure strength programs, but it can also be a break for your CNS to recover which might be the limiter right now if you've been at it for a while. Is your goal maximum strength with little regard to size, or are size and strength equally important? If it's the latter, I would give a cycle like HST a try.