Hi,
> When I use the turn signal stalk (in either direction) or activate the
> hazard lights the indicators (turn signals) blink one time. On the second
> blink its like they don't have enough juice, the dash signals glow and go
> out. No more blinking.
>
> Originally I thought it was a broken turn stalk (as I was driving when this
> happened), then I checked the hazards. All the bulbs work (activating the > alarm).
That sounds like the turn signal relay. You will find it directly behind the red hazard lights triangle in the middle of the dash. In fact, the red triangle is the front side of the relay and not easily separate from it.
Even if the hazard lights themselves blink properly, the turn signals may not work. That's what happened to me in my 2001 A4 Avant (also a B5 chassis like yours).
In Europe it costs approx USD 50 to replace. Remove some of the black plastic panel around it and lever it out carefully. What keeps it in place is the friction from a large number (6? 7? I don't remember) of spade connectors.
/Robert