what is the correct tire size for a trek 7300 multitrack?
Bike Blue book has the original tires on a 2012 model as 700c x 35:
 bluebook

You could just go with any 700c x 35 tire. I've found that many local bike shops can order for a similar price as mail order and they'd be able to match your tread to the terrain and weather conditions in your area. As far as width, you might be able to go slightly wider with a 37mm (as you currently have) or 38. It all depends on your tire clearance, especially if you have fenders. 



As for your particular tire, I agree the Slickasaurus have ambiguous markings. The manufacturer's website doesn't give any information so I took this screen grab off Amazon for a Slickasaurus tire that shows  "700 x 35c (35/32)".  My reading of this is that it fits on 700c rims normally designed for a 35mm tire width (that is, the wire bead is close to that of a 35mm), but because it's a slick tire, the actual tread width is closer to what a 32mm tire would have.

For your old tire, it means that it nominally fits rims designed for tires that have a wire bead profile closer to a 37mm tire's wire bead but in actuality has a tread profile closer to a 32mm profile.