"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."
I'd check out the ground on the right rear signal/brake bulb. As a test, touch a piece of wire to the bulb base near where it meets the glass. Touch the other end of the test wire to a known good ground. If the bulb starts acting normal again, it's a bad ground to the bulb - bad ground wire, bad socket, poor connection.
I'd check out the ground on the right rear signal/brake bulb. As a test, touch a piece of wire to the bulb base near where it meets the glass. Touch the other end of the test wire to a known good ground. If the bulb starts acting normal again, it's a bad ground to the bulb - bad ground wire, bad socket, poor connection.