That 28-inch designation is a European equivalent to 700c. Both have 622mm bead diameters. In the States, just search for 700c tires in an appropriate width.

According to Jamis, your bike came with Vittoria Adventure tires. Those are a bit cumbersome (700-800g) for recreational riding. If you ride paved and smoothed unpaved surfaces and want to get away from clunky tires, I'll recommend taking a look at Panaracer. I have over 15k miles on Pasela TG tires, and they've been great overall. The Panaracer T-serv is the very close cousin to the Pasela TG, but it has black sidewalls (rather than the gumwalls on the Pasela) that are supposedly more abrasion resistant. Affordable, good puncture resistance, reasonably durable and very comfortable. A set of 32mm T-servs will be about 1-1/2 lbs lighter than your 35mm Adventures.

Vittoria makes several tires in their "Randonneur" label. I believe the current offerings include the Randonneur Pro, Randonneur Hyper and Randonneur Cross Pro. I haven't ridden any, but lots of people speak well of the first two.

Incidentally, while these tires aren't slicks, don't let the shallow tread deter you. It's next to non-existent and won't be functionally different than a "slick."