I’ve got WTB Exposure 30c as my ‘road’ tyres on my second set of wheels, which have done a little bit of light (and dry) off road … The tread is lightly textured on the shoulders but not sure it gives any great advantage off road. They’re good enough tyres, especially as I got mine, lightly used, for £15 the pair!
If you’re after cheap, I’ve just put a set of these on the commuter …
https://www.planetx.co.uk/i/q/TYCONSCII/continental-sport-contact-ii-700c-wired-tyre
Listed as 32, 37 (they’re not 37, they’re 35, at least the ones PX sent me … get it sorted @brant!) and 42. Roll pretty well and a little bit more capable off road, although will get overwhelmed by anything approaching mud) But at £8/10, you can’t really argue for the kind of use you’re suggesting …