People are out touring on almost any rig imaginable. It's possible to do a tour on a performance road bike. It's possible to tour with a mountain bike. It's even possible to use a cheap department store bike (although I wouldn't recommend that option.) There's a lot of flexibility in bike touring.

If you've got a road bike you like, keep using it.

If you can't add a conventional rear rack, there are some racks which attach to the seat post. If you go this route, you need to keep your gear load relatively light, but it's still much better than carrying the load on your back.

If your bike cannot accommodate any racks or if you don't think it could handle the extra weight load, go with a small trailer. Burley makes some great trailers and BOB has a solid reputation too. A trailer will be a lot cheaper than getting another bike. However, before you go this route, do some research and see if you can test what you want to use.