Assuming you've got the colors/functions correct, it doesn't make much sense. If connecting the running light wire (brown) causes other lights to go out, the new lights must be pulling that line low. But if that's the case, I would expect a fuse to blow.
What brand/model are the new LED lights? It sounds like you got it right, since yellow(and green)=stop/turn, brown=running are common colors and if all that's left is black, it should be ground.
(Switched ground would be a real oddball. The signals from the TV use a common ground with all common trailer connectors, and most taillights are wired for that.)
