There are no mechanical blinker/flashers on a late model Mustangs. It's handled in a solid state electronic flasher module similar to the generic electronics module (GEM). Because it is solid state it rarely fails. However, the fact that the hazards flash at all suggest the module is working at some level.
The manual lists the Electronic flasher module as "behind the dash panel".
DOUBLE check fuse F2.18 AND F2.13 in the Central Junction Box (CJB). Don't just look at it them. Pull and check with a VOM. Next check the ground near the center console. The ground is shared by many devices including the radio. Sometimes during a radio install, this ground is not re-connected as it should be.
If the fuses are good, most likely your problem is in the multi-function switch (specifically in the hazard switch itself). The multi-function switch is located on the steering column. The switch can be rebuilt. If you want to rebuild, pay attention to the location of all springs and such (may help to take pictures). The inside is filled with di-electric grease. Clean the old grease. Lightly sand the contacts. Carefully bend to make good contact. Replace the di-electric grease (OK to use the same kind of silcone used for spark plugs). Re-assemble and test.
An option is to get another multi-function switch from a salvage yard. The Crown Vic/Grand Marquis from the same year are interchangable (actual interchange is wider but the stalks will appear different). Rebuild the replacement. Then if something bad happens, you are not risking your DD. Also, once rebuilt, it will reduce the time your DD is actually down.
The original can then be rebuilt as a spare.