Closer to home and south of you, try Roper State Park near Safford. If you prefer to eschew egalitarianism, try Essence of Tranquility also in Safford, for a spa experience.
Just north of you is Verde Hot Springs, between Verde and Payson. Several miles of dirt road in -- check weather conditions. The old spa/hotel is abandoned, but the baths and pools are still in good shape. Right on the Verde River, which usually has a good flow.
West of you in Tonopah are the El Dorado Hot Springs. Spa or "self guided".
Or, (can't help myself) wait 'til June and buy a kiddy-pool and two cases of Evian.
Fair Winds and Following Seas