I think I have to agree with DennisM, while you may have some grounding issues, this sounds more like a routing issue.
A couple of suggestions,
1)If you have the wires crossed at each tail light (I would assume the same for the park lights) you will end up with a weak tail light and a strong blinker. It has to do with the specific filament in the bulb used for either the blinker or tail/brake (or something along those lines). I know because I had this condition myself (it is easy to cross these if you are not using a stock harness).
2)Are you sure that you have the correct wires hooked up under the dash? You state that both tail lights are flashing. There is a separator coming out of the turn signals that send the flasher to the specific side yet still maintains the brake light being available at any time so you can check this connection and make sure you are correct. Even a non-stock harness should follow the turn signal/brake wire routing as shown in the shop manuals.
3)You state you are using a non-stock wiring harness (nothing wrong with that - i am doing it myself), so it may have come with a rear flasher capability (i don't recall that as an option on these trucks) and that may be the culprit if you are trying to have those function on your truck. Yes it can be done, but you have to have the wires routed correctly.
Finally, if you are concerned with a ground, pull the bulb out, run a specific wire (outside the truck) from the bulb directly to the negative battery post and put a trouble light in between, if you get a good light and you don't without the dedicated wire, yes you have a bad ground.
