Sport touring "mixte" - need help choosing tires

[warneral]

I just got a cheap Ross Gran Tour bike (early 80's - heavy double butted 024 Ishiwata steel). It is a sport touring bike - mixte (single top tube) I don't know why I did it. I really want a better bike for tugging my 6yo on her tagalong I thought this would be nicer than my heavy upright navigator.

The bike is the same as this guy talks about mid way through:
http://bsanders.net/sporttouring/

I'm already kicking myself for bringing it home LOL
.
The only thing it *really* needs is new tires & tubes. The original tires are still on the bike. they are 700 x 28c

I was considering trying cyclocross tires? Something that would work on crushed limestone when we ride on paths. Is that a bad idea? There definitely looks to be plenty of clearance. Do they make cross tires in that size for a decent price? Could I buy wider tires for these rims or do I need to stick with 28?

Thanks!!