I think you're on the right track here. If moving the bulb just a bit makes a difference, there's a bad contact. Check inside the sockets for corrosion on the metal parts, and down at the bottom, check that the contacts (should be two for the combination tail/stop light bulbs) are clean. (It could also be a problem where the wires connect to those two contacts in the base of the socket, but less likely.)I found that the bulbs were dandy fine, but were VERY touchy in the sockets. In other words, they would work fine if the bulb was JUST right in the socket . . . . . Could it be my sockets or just a bad run of bulbs?
The fact that both new tail/stop light bulbs seem to have the same problem, it sort of points to the bulbs, but if they're the right type, this shouldn't happen. My 07 Owners Manual doesn't indicate the actual bulb model/number, just the Wattage of the two filaments (21/5). However, the original bulbs should have the type number impressed on the metal base.