I ended up getting new tyres for the bike… I thought about getting airless tyres, but in the end just got random tyres from Mike’s Bikes who came highly recommended on a mailing list at work, and seemed really nice.
I got Specialised tyres randomly, and they’re really good. I can’t remember their name now (I’ll look it up later), but they have a tall center section which makes them look and act like road tyres on hard surfaces, and a knobbly bit on the outer edges for loose surfaces. Overall, lower rolling resistance on the hard stuff, and they gripped really well on the soft.
I like them a lot.
Update: So, the tyres are Specialized 06 Crossroads, or at least that’s what the packaging says.