Is it open at the Spence Hot Springs right now and where is closer to camp from to get there?

I know you asked and got a half answer 11 months ago but even in a "pandemic" they do not ban dispersed camping. You should look up the rule book for dispersed camping but there are no locations for it and they are everywhere outside of city limits except for private property. They are not camp grounds and they are legal up to 14 days in New Mexico.

No overnight camping permitted within the State, currently. Yes, Spence is open.