Well I finally got around to installing the new turn signal switch. Unfortunately, the new switch did not fix all the issues. At least I have a working hazard switch again and I no longer have the issue of jiggling the switch causing the lights to cut in and out. The braking problem still remains.
To recap:
(1) When executing a right hand turn there are two scenarios that have the same outcome.
(a) I brake and then turn on the right turn signal. This causes the right to flash, left turns off.
(b) I turn the right signal on and then brake. This causes the right to flash, left never turns on.
(2) The opposite side does not act this way. Left turns behave as expected.
Back to the drawing board I guess.