Errrr, first I think we need to clarify some terminology here.

700c and 650c refer to WHEEL (or rim) size(diameter), not tire width. 700c is the most common for road bikes, although you will see 650s on smaller (women's) bikes and some tri-specific bikes. Wheel size has nothing to do with flats.

Tires come sized for both 700 and 650 wheels, in a range of widths. It sounds like your bike shop switched your standard(for road bikes) 23's (that number referes to width) for some wider, beefier 28s. 700X28s are great tires for touring, tandems and hybrids, so you should be able to find a fairly wide selection. Personally, I don't think that tire width (or narrowness) causes flats either. Usually when a rider comes in with "terminal flat-tire-itis" it is due to tires being run at too-low pressure.

As far as fitting tires in your bike frame...I do see certain frames (of all sizes) that will only accept certain tire-sizes. Without seeing your bike, it seems to me that your bike already has pretty good clearance if you were able to stuff 28s in there! I *think* (and I could be wrong) is that your bike shop is saying you won't be able to stuff 28s with slight knobs or tread (aka "semislicks") in your frame. So here is the tricky part...finding 700c tires in a narrower size with knobs/tread. Most 700X23 (or 700X25) are slick road tires.

The fender clearance issue a whole new can of worms......

So while I was typing this I started to think that perhaps the easiest, best, solution for you would be to start looking for a winter beater/commuter. An old mtb would be ideal and it will save your nice road bike from getting destroyed (it's steel if I'm not mistaken) by the water and salt on the road.

Sorry, I couldn't be of more help ...check out icebike for more winter riding/commuting advice.