Minor electrical problem; possibly a bad ground

I've had this minor problem for a while, but it only happens when I drive at night with the lights on, which I rarely do, so I haven't fixed it yet. 69 Barracuda FB. Both front turn signals work fine (nice and bright) when the headlights are off. However, when the headlights are on, there are two things wrong:
1. When I turn on the headlights, without using the turn signal, the right green turn signal light on the dash illuminates, and the right front turn signal/running light will does not come on. If I tap it, it will flicker. Sounds like a bad ground, right? But that light bulb works fine as a turn signal when the headlights are off.
2. With the headlights on, the right front turn signal will blink, but it is not nearly as bright as when the headlights are off.
Rear turn signals work fine. I was thinking that I might have a ground problem. But if that was the case, why would the right front turn signal work fine with the headlights off? But when the headlights are on, and the right front turn signal/running light does not come on, but it will flicker when I tap it makes me think about a bad ground. So somehow the headlights being on is causing my problem. This is really irritating, since I am usually pretty good with electrical problems. Not as good as @67Dart273, of course, but pretty good.