I've cooked some great ribs, pulled pork, and other slow cooked meats with a propane grill. It isn't ideal, but there are a couple of tricks:
1) Just because there is a burner, doesn't mean you have to turn it on. I will put one burner on the low heat setting, and drop a pork butt or ribs on the cooler side with no direct heat. This allows the chunks of meat to cook slower without direct heat.
2) Use a water pan to act as a thermal buffer. Use a low profile foil pan, fill it with water, and put it under the grate and over the burner. The mass of water absorbs the intense direct heat, and prevents burnt edges. It also keeps the humidity up so everything stays nice and moist.
3) I use dry wood chips wrapped in a foil pouch with tooth pick holes poked into it for air flow. I put them under the grate and near the burner so it starts to smoke.
4) Get a good grill thermometer near the grate level so you can adjust the burners to an actual temperature. I am playing games with a non-uniform temperature gradient to get the temperature where I want. The average grill has a thermometer that is 6 inches above where the meat is. I use a digital thermometer that has two probes, one that snaps into a holder on the grate to monitor the temperature close the meat, and one that sticks into the chunk of meat. A good thermometer makes cooking so much more consistent for me.