The guys have given you solid advice above. You need to let them know if the electrical system on the car is stock or not and if any aftermarket gear has been added to it.
1) If this problem all of a sudden started then I would lean towards a resistance issue as 7991LXnSHO stated. You should check all major battery cables for connection corrosion or just being loose. For example, I literally had a 4 ga ground cable from the battery down to the K-Member internally oxidize to the point of causing the car hot start issues. Also, check the headlight harness grounds, should be one at each corner of the core support on the top near the fenders. Should have a green 7mm or 9/32" hex head on the bolt that secures the ground wire to the core support.
2) Have you added any electrical load to the car? For example, electric fan(s), MSD box, stereo, etc. I believe the stock alternator is a 75 amp unit and cannot support much more than what the car came with. If you are adding electrical load you really need to start thinking about a 3G conversion if it has not been done already. If you want to do this go here and scroll down to the "Electrical" section and there are two writeups on how to do it.
Guys, if "that should be a sticky," then please reply to this thread with the link to your suggestion and why you think it should be added. How to determine if a thread should be included: - Does the topic address a common fox 5.0 owner's needs? / Is it a frequently asked question? / Is it...
3) LED headlights - not much experience here but all I can say is do the research and your focus should be on beam pattern and having a clear cutoff.
4) If the headlight bulbs are not the stock units i.e. they were replaced with something hotter like the Sylvania Silverstars (this is what I run) you should look at putting relays in so the headlight switch stays cool. Do a search and you will see what I am talking about. This also made my headlights brighter as it increased the voltage at the bulb. You can wire this yourself (what I did) or if you do not want to put the parts together you can get kits:
Detailed technical information: Fix dim headlamps!
Again, I would start with making sure the battery is in good shape (have it load tested), making sure the alternator is is top shape, checking the battery cables (positive and ground both) and then looking at the headlight harness grounds in that order.