My driver side headlight bulb keeps going out. What can I do? ?

Look for loose mounting, bad/ loose  connections. Especially ground. A tungsten bulb  draws10x its running current on starting, so if  it is being "switched" off-on-off   a lot by a loose connection,  it's cycled  more than it should, and wont last.

   Any knuckle-dragger who cant tell :"shorts"   From Opens has  No  business with electricity.. Shorts Blow Fuses...no blown fuse was indicated..

   If a bulb melts a socket. get an  LED. I had to.

check the light's mounts for looseness, vibration can cause early failure

There is likely a short in the circuit and it is taking out the bulb.  Time to see a good mechanic.

My wife had a 2012 Hyundai that ate front bulbs like crazy and one of the taillights melted the socket twice.

Bulbs burn out and have to be replaced.  Some sooner than others.  I am wondering what the problem is that you had to replace the wires.   I think that is where your real problem is.  Are you sure that you are installing the correct bulb? Because I am imagining a lack of resistance that is melting the insulation off the wires.

Time to take it dealer for deep diagnosis, it will cost you