2000 Oldsmobile Bravada combination switch

That's not enough information to reach a diagnosis, but when you add that a wrong light turns on with the intended one, there are two things to consider. Most commonly, check for a trailer wiring harness that is chewed up, often from rubbing on the ground when you go up a steep, sudden incline like a driveway entrance. This was also a real common problem caused in the '90s by a lot of trailer harnesses installed by U-Haul. They have three red leds to help you identify the circuits. I ran into a bunch of them with connectors that were melted and the leds were shorted.

This would also happen when an 1156 bulb with a single terminal was installed in place of the required 1157 bulb with two contacts. The single contact on the bulb straddled the two contacts in the socket, shorting the turn signal / brake light circuit to the tail light circuit. You should have the newer 3157 bulbs with the plastic bases, so this can't happen, and you have two separate bulbs for brakes and turn signals. Where you still can have a problem is with a bad ground wire that both sockets share. Current for one bulb can't get to ground directly to cause the bulb to be full brightness. Instead, it finds an alternate path through a different bulb, then over to that bulb on the other side of the vehicle, then to ground. Now you have the 12 volts split up between three bulbs instead of one so while they all might appear to be on, they will not be as bright as normal.

That condition of three bulbs in series when there's a bad ground also results in greatly decreased current flow. With older mechanical flashers, that current is insufficient to make it work, so it won't flash. Newer cars have electronic flashers that flash too fast to let you know there isn't enough current flow, which is normally caused by a burned-out signal bulb.

The first thing you have to explain is whether this problem started after the multifunction switch was replaced, or if you replaced the switch to try to solve this problem.

We used to have a real simple and reliable mechanical signal flasher that cost around three bucks, but the switching between using a rear bulb for a signal light or a brake light was done with a rather complicated turn signal switch. On newer vehicles they went to a much-simplified turn signal switch, and built it into a complicated multifunction switch assembly, and went to an electronic flasher worth around 50 bucks. I've only looked at a few of those, but they always had two relays in them, one for each side of the vehicle, and lots of electronic circuitry to run them.

I would suspect that electronic flasher based solely on your observation that the system seems to work properly for a few minutes, then the problems occur. "Crossed wires", meaning two that are rubbed through and touching will either act up randomly or all the time, not consistently like you described. "Crossed wires" can also mean two that were removed from a connector and reinstalled in the wrong places. Wires don't jump out and switch places by themselves. You would have to have done that accidentally yourself, and then too, the problem would be there all the time.

Electronic circuitry is famous for causing weird and intermittent problems, and starting to act up after a consistent amount of time is one of the characteristics we typically find. Try plugging in a different flasher first before we go looking for a less-common, more elusive cause.