I don't see how the turn signal switch could affect the headlights - two totally different circuits and independent from each other.
I'm betting that you have a separate relay in there just for flashing the high beams (if you pull the lever you'll hear it clicking so you can find it) and all other functions (headlight, taillights, etc) work off of the headlight switch and then the hi/lo relay.
I doubt that the hi/lo relay is bad, as that doesn't affect the tail lights, so the only thing that affects BOTH head and tail lights is the headlight switch, the feed wire running from it to the fuse block and the fuse in the block (of course, all wires and slip-on connections between the battery and the headlight switch are suspect, too).
JPS is really a VS, and they've used a VW fuse block (I believe - if so, it should have small ceramic fuse bodies, pointed on each end) and, if they wire it the same as a VW, it should have separate fuses for hi and low beam on each headlight, and separate fuses for left and right tail lights.
Let's do this one step at a time: get a voltmeter, set it to the 25 volt range and probe around the headlight switch to see what's going on. I believe that (you guys always catch me with my service manuals 1000 miles away) the incoming power to the headlight switch is on terminal 30, but there should be TWO terminal 30's - one coming from the voltage regulator, and another going to the turn signal relay. BOTH should have 12 volts at all times. The tail lights are on 58, and the headlight relay on 56 (it may be the other way around - you're really pushing a 57 year old brain here). They should show 12 volts when the headlight switch is pulled out to the first and second positions.
Probe around and report back what you find.
BTW: You asked if you'll need another relay for those flame thrower headlight bulbs.....yes, I'm afraid so. The VW relay is good for something like 15 amps on a good day and you're asking for it to manage twice that amount. You'll need another, 30 amp, headlight relay just to run the headlights. You need a "single pole double throw" relay (get one for fog or driving lights at some place like Autozone) and you can make it switch (energize) off of the Hi-beam lead from your existing headlight relay (terminal 56a, I believe, but it might be terminal "f" - whichever is hi beam). Once installed correctly, the load on your existing headlight switch and relay will drop to less than one amp and those two components will last longer than you will.
Again, probe the headlight switch and report back.......
Tx........gn