Since junkies constantly re-use needles I suspect that they can be re-used but I would think that any hospital regardless of size etc would be doubly careful not to make any mistakes if the hiv wing was in the same area.
After all forget the poor old patients - the staff would take extra care for their own desire not to get infected!! That may in fact be your best reason to relax. Self-interest is pretty high in most of us & add their fear of not wanting to catch anything, to their fear of being sued, losing their job or of being charged with say "criminal neglect".
No idea what kind of needle they'd use but I really don't think that would be the problem - if there was a problem & as above I really think that is unlikely - but if there was it would be human error & I still think pretty hard to happen.
As to the timing of your cold - unless you were quarantined for that week you were in contact with dozens maybe hundreds of people who could have sneezed etc & hey presto you get a sore throat.