Yeah, you want a lever action rifle. A single shot won't work.
The SBH is fine. You will have to shoot in the Modern category because of the sights but that's no big deal. Most folks use BHs with non-adjustable sights. If you have been out of what's new in the world of Ruger, they are called Vaqueros but they are still Blackhawks of sorts. You can shoot one of each but will still be in Modern. Yes, you do need two.
No, any double will not work. It cannot have ejectors. Extractors yes but not auto-ejectors. You may be able to disable them on yours if you have them.
The good news: go to a shoot. Take what you have. Take ammo (about 200 rounds of "cowboy loads"). Take 12 gauge shells (2 boxes of AA featherlites). You will be bombarded with guys wanting to loan you or share guns with you. Some guys take spare leather and clothes for just such occasions. You can shoot what you have and borrow the rest. Don't wait until you have all yours guns, leather and outfit. If you do it that way here's what you'll be buying if you have nothing now:
2-Ruger Vaqueros, Colt SAAs or clones
1-Henry copy, 66, 73, 92 or 94 Winchester copy or original
1-SxS double trigger, non -ejector model with internal or external hammers or a Winchester 97 or clone
Gunbelt and two cowboy holsters
Shotgun belt or shotshell slide
Boots
Pants
Suspenders
Shirt
Vest
Scarf
Hat
Spurs & straps
Pocket watch and chain
Chaps
and so on.
You could get one more SBH ($300?), a Rossi 92 clone ($250) and an import SxS ($200) and go shoot in work boots and a cowboy hat in jeans and a long sleeve shirt you already have. The only thing frowned on, even for beginners, is wearing "designer" jeans and shirts with logos.
Again, go to a shoot. Most clubs have a for sale area and you can get great deals on the spot from club shooters. You can spend as much or as little as you want. If another new shooter shows up you could even partner with him. Your pistol and his pistol, you buy a rifle and he buys a shotgun. Share from now on. I don't mean to talk down to you. You may be a rich record exec and go out and buy all new high end stuff tomorrow. Again, go to a shoot. It will be worth the trip even if you only shoot a stage. This game is hard to explain until you see it first hand.