Okay, you're dealing with two different issues here and what I'm about to tell you is going to sound illogical. Trust me when I say this, I have a ton of experience with this car and the problems that you are experiencing are not only common with this model, but I have personally fixed the same exact problems on mine. As a matter of fact, I'm working on my air conditioning right now:
Yea, it ain't pretty or fun.

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Here's the deal with your headlights. You have a headlight motor gear that is going bad. You won't know which one it is until it actually goes and when it does,
this is the sound that you will hear once you turn off the lights and they go down. At that point, you will know exactly which motor has the bad gear. It is very loud and noticeable and the sound makes your car look like it's crap.
Here's a writeup that I did of how to fix the problem.
Now for your turn signals. The problem is most likely the hazard switch. This is also a well known problem. The way that the turn signal circuit is designed, routes it through the hazard switch. The switches either wear out or the contacts on the switch gets dirty and cause it to work intermittently. I would say that this is your exact problem but the running lights that you are having a problem with throws a monkey wrench into the equation. That makes me think that your turn signal sockets may be shot due to water intrusion. Check out this picture.
That picture is the passenger's side headlight well with the headlight removed. If you can look at your housing and see a hole in there like what you see in my picture, you may need to change your housings and the LS15 socket that holds the turn signal bulbs. Warning, those little sockets are NOT cheap. The lenses are cheaper.
One thing that you can try is this. Press the hazard switch a bunch of times very rapidly. Then try your turn signals. If they start working, the hazard switch is bad. If they don't, then you have a wiring issue. Lucky for you, I have
a schematic that you can look at.
Let me know what you find out and we'll take it from there.