Unfortunately there's no overlapping version of Unleashed which would run the R310 and R350.
You could buy some R320s (which are the same generation as the R510), and they'd happily run Unleashed 200.7.
Or you could accept that you're running a geriatric Unleashed network, and leave it that way: just buy 3 (or more if you want spares) used R310/R500/R600s and keep the money you saved for a real upgrade later.
Or you could buy the R350s + a ZoneDirector 1200 (which is nearing end-of-sale). Used ZD1200s are $150-200 off eBay with enough licenses for your site. You'd have to re-setup your network configuration and update all your APs with the new firmware, but you'll be able to run all of your existing APs + the R350s on the very latest 10.5.1 release.