IF the winker lights light up WITH the switch, but don't blink... I'd be looking at the winker canister... The handle bar control only sends power to the winker relay and related system... 
  The high beam  indicator light bulb should be on the same short bits of wire that FEED the high beam... I believe mine is the yellow wires.
  Some times electric issues can be frustrating.. Patience and a bit more checking... It always seems when you are stuck and someone you know, understands this stuff, can walk right up and say.. This was hooked up backwards... or feeding the relay the wrong way..
Sneezles61 

Thank you, but as I said...the flashers won't blink with the original flasher relay installed OR with the brand new relay installed.  As for the high beam indicator, I'll have a look at that yellow wire that feeds it. 

Are you trying the turn signals with the motor running with some RPMs? I think I asked that before but don’t recall if you answered…sorry if I missed it. On my RT3 they light up but don’t flash until motor running and up on RPMs. Probably sensitive to voltage. You may not have any troubles at all. Try it running as I suggested. Hopefully you’ll report back with thumbs up!

Pedlacrazy...the turn signals won't flash with OR without the engine running. I wish the solution were as simple as you had hoped.  Still hoping for an obvious "DOH!" wiring/connection issue that I might find, but right now, everything looks to be connected the way it is supposed to be.

You either have flasher connected backwards or got a bad flasher.

I connected the new flasher relay exactly the way the seller (Pobit) instructed.  The instructions he included are shown below. 
 

Here is a better image of Pobit's instructions: 

Here is a better image of Pobit's instructions for his new flasher relay: 

Some flashers don't care, many do. What are the terminal markings on new flasher?

I don’t have much experience with flasher relays but I think it should be fairly easy to test it off of the bike. This way you can know for sure if it works and help you zero in on the source of the problem. I found a video on you tube where a guy bench tested a flasher relay with a battery and a test light. Seems pretty straight forward. Good luck.