'99 Mustang Hazards work but turn signals do want

Multi-function switch

Assuming all fuses are good, IMO this is most likely the multi-function switch.

It is possible to rebuild it. If you attempt it, take pictures as you go. There are tons of springs and cams that are not that difficult but can be messed up.

Clean the entire switch with electrical contact cleaner. Lightly sand all of the electrical contacts. Re-grease the entire switch with di-electric silcone grease (same kind as used for spark plugs).

Another option is to get a replacement from a salvage yard. Try car-part.com. Rebuild the salvage yard unit. That way if you make a mistake, it won't affect your DD.

Remember, that Ford uses a similar multi-function switch in other models. The replacement can come from a Crown Vic or Grand Marquis. The stalk may be different but it will work. Be sure to bring the Mustang unit to compare.

Consider writting a "how to" thread.