Another quick update with nearly a complete fix...
So, I yanked both bulbs out tonight, took them apart, and soldered in a diode before the main resistor to ensure electricity only flows one way. Slapped them both back in and clicked on the blinkers... left blinker on, only left bulb blinked, hooray! Right blinker on, both lights blinked...WTF?! Long story short, one of the diodes I soldered in was bad (what are the odds?!?!). Swapped it out and now only the directed bulb blinks when commanded, hooray! But, I don't think I am all the way out of the woods yet.
With the headlights on, I'm not sure its doing everything correct (hopefully somebody can confirm correct light function). With headlights on, taillights are on, front running lights are off. I then click the left blinker on and it blinks from high to low brightness (vs high to off as would be expected). The dash turn signal indicator bulb does the same thing; high to low brightness vs on and off. I don't know what to look at for this issue, so suggestions are welcome.