I can't find the original posting on Shrewsbury, so here is my answer:
It depends
The posh way is Shrowsbry (rhymes with throw)
- this is perceived by non locals to be correct.

Many of the locals say Shrewsbree (as you said)

but where I lived - up the road a few miles - they said
Shoe-s-bry as in shoe (search me as to why)

So it doesn't matter how you say it, you just need the evidence to prove that you are right!