Well Bicton is really a village/suburb and as far as I'm aware there are no apartments there. You must have seen retirement flats? There might be some maisonettes.
As for Shrewsbury
town centre - it's hardly Manchester city centre. Most of the new apartment developments are in quiet areas. One exception that springs to mind is Nexus on Roushill, but that's quiet compared to much of Manchester!
Some apartments you might want to consider looking into are at Blackfriars Crescent (St Mary's Water Lane), Benbow Quay (Chester Street) and the Old Brewery in Coleham. Have a look at the Nexus on Roushill too - you get parking in the town centre there.
Here are the Pooks (one of the larger estate agents) residential properties available to rent -
http://www.pooks.co.uk/iframesearch.htmlSome apartments are above shops on main town centre streets. Others are down side streets and passages (which in Shrewsbury is a good thing - quite the opposite to Manchester in this respect). It's worth having a look to see exactly what and where an apartment is. If you find somewhere in the town centre but aren't sure, just ask here.