Rear axle keeps developing play, is it fixable or time for a new wheel?

IMHO, Time for a new hub. Whether you get a full new bike...well that would depend on what you have currently, and the kind of riding you do combined with how much you want to spend. The loose bearings and freewheel just don't do well once they are worn.

It is worth noting however that getting the cone tension just right takes practice, you have to do it several times to get a feel for it. If you have the time to mess with it, you might be able to pull it off.

Over the summer, my son and I volunteered with new bike assembly at our local non-profit bike shop. The hubs that were brand new, often came overtightened, and were quite a pain in the butt to get just right. My 13yo son actually has a better knack for it than I do. You have to overtighten the cone just the right amount so that when you snug it up with the lock nut, it loosens to the right amount of tension.

Re "cones are properly adjusted". So you set the bearing tension, and there is no play until you ride on it a bit?...or is it that you can't seem to get the right tension. Too loose is sloppy, and too tight is crunchy?

The practical amount of money to spend on a new hub is of course dependent on the bike, and what you want to do with it, and building a new wheel may not be practical for you. If the bike is worth $100, spending $100 on a wheel does not make a whole lot of sense. Tell us about more about your bike, and what kind of riding you intend to do.

Good luck, Ben