Without a doubt inadequate grounding can cause strange thing to happen. Like suggested check to make sure all, rear and front have good grounds. On 69s the socket grounds directly to the metal reflector, lator years used a black ground wire from the socket to a grounding screw. Also make sure the light sockets are free of corrosion.
It’s possible you have another problem. I don’t understand what lights are on/off when you say.
“but when I have the right turn signal blinking and hit the brake, only the right turn signal stays on until I let off the brake where it begins blinking again and the brake light on the right comes on again.”
With the brake on does the right turn signal go on constant? Is the right front signal light off or on? is the left brake light on or off?
“If I have the hazards on and hit the brake all the lights, including the front turn signals just stay on”
This is normal. Unlike the turn signal switch, the hazard light switch doesn’t disconnect the brake light circuit from the brake/signal lights.
It’s possible you have another problem. I don’t understand what lights are on/off when you say.
“but when I have the right turn signal blinking and hit the brake, only the right turn signal stays on until I let off the brake where it begins blinking again and the brake light on the right comes on again.”
With the brake on does the right turn signal go on constant? Is the right front signal light off or on? is the left brake light on or off?
“If I have the hazards on and hit the brake all the lights, including the front turn signals just stay on”
This is normal. Unlike the turn signal switch, the hazard light switch doesn’t disconnect the brake light circuit from the brake/signal lights.
