JF,
If you have NOT done anything recently (messing with wiring) and the stop filament was working in those modes, then recently stopped, I agree it may be the ground. "The ground" is a big word, there are several places that could be breaking down due to loose fit, corrosion, rust.....First place to check is the bulb base to socket. It always seems strange about ground. At first thought you would say, if the thing works when lights are on, then it is getting ground. But there is an explanation for that which we don't need to discuss yet. It is also strange that both rear lights have this problem. Their grounds are separate. This leads me away from ground. But "it's always the ground"
Just like "it's always the battery".
Let us know if the tail light flashes (turn signal) during the following modes.
1. Lights on
2. Lights off
3. Lights on brake on
4. Lights off brake on
5. Does the dome light work from headlight switch?
This info may help us in troubleshooting.
The turn signal filament is the same as the brake filament. FYI: The brake switch gets it's hot feed from a dedicated fuse. When brake is applied, switch closes and sends power to the turn signal switch, which routes the power to the rear tail light brake filament. The reason is it routed thru the turn signal switch is that brake power to the rear is shut off at whatever turn signal needs to flash on a turn. Otherwise that bulb/filament would not flash. This may help in your examination of the circuit, if ground is not the issue.
It is better for a forum if a person starts a new post when a different question is asked. That way you get your own subject heading and folks will see it.