Being able to move the top of the valve stem around tells me you've got a badly worn valve guide. That's a major repair that involves removing the head and getting a machine shop to rebuild it. If your extra head has a casting number that ends in "848" that's the correct one for a 1957-62 engine. How about getting that head rebuilt since it's already off, and swapping the rocker arm assembly to it when you install the rebuilt head? That will let you keep running the present engine while the replacement head is getting rebuilt.

Don't let a machine shop talk you into installing hard valve seats. It's not necessary unless you're planning to run a "dry" fuel like LP or compressed natural gas. The stovebolt six exhaust valves are identical to the ones used on a Chevy small block V8, and stainless steel ones are available dirt cheap in sets of 8 on Ebay. You'll spend much less than buying six "numbers correct" valves for a stovebolt engine.
Jerry

"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
Cringe and wail in fear, Eloi- - - - -we Morlocks are on the hunt!
There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. - Ernest Hemingway
Love your enemies and drive 'em nuts!