So, it is P0344, not P3044.
First, check your cam position sensor connector and wiring on the passenger side of the engine. The cam position sensor plugs into the OPDA assembly that plugs into the side of the engine near the oil filter. The sensor connector plugs into the wiring harness and the wires then go over the dipstick bracket and sometimes folks have issues with the sharp edge wearing through the wire loom.
Next, if the wiring looks OK and the sensor connector is not corroded and looks good, then I'd change out the camshaft sensor for a new Mopar one and see if that helps. As you are looking into that, and since you have an 05 TJ, do some searching and reading on the OPDA and P0344 code etc. You should probably determine if it has been replaced before on your TJ and if your TJ falls under the 05 recall. You may want to replace your OPDA with a Crown aftermarket unit if it hasn't been done already. This is a long, complex, and sometimes dark journey for some unfortunately.
I also have an 05 RME (not in the recall), and have some experience with the OPDA already. It's a stupid design change they made on the 05/06 TJ's. The OPDA needs to be maintained as a regular wear item. If the bearing on the OPDA shaft goes bad and the OPDA runs off center, the OPDA gear can cause wear on the camshaft gear threads that are complementary. Once that physical gearing union is not stable, after major wear has taken place, you are looking at a camshaft replace, new OPDA etc. Worse case scenario, since the OPDA drives the oil pump and if it fails catastrophically, the engine can be starved of oil and seize up. Usually it isn't that bad though. But you can see why it is a major concern for us 05/06 owners. I've replaced my OPDA with the Crown unit after battling what you are describing. So far, so good...my fingers are crossed! Don't mean to scare, but I just wanted to give you the straight up Cliff's Notes about the OPDA.