The front turn signals do not work. The socket for the bulbs are appear to have got too hot and are melted. Can I replace the socket by getting another socket and splicing, or do I have to replace the whole wiring for the turn signals.

Dear Alex,
The problem is in the bulb, or the socket, or the wiring to these.
Whenever a turn signal flashes fast, know that one of the bulbs is not working.
You have already identified the problem as being the front one. Problem could be the hot wire, or the ground wire, or the ground connection to this bulb.
Start by taking a tester to the hot wire, and see if it is receiving electrical impulses when the turn signal is on.
( It probably is, unless the car has been in a wreck and has been improperly repaired.)
Then check the socket or the wires to is and see if the bulb is grounded.
I suspect the ground is defective, and either a new socket or a repair to the ground wire should fix you rights up.
God bless your efforts.

If bulb or socket melted then replace socket. May be OEM part not sure. Easy fix just be sure to use seal type crimp connectors or you will be working on this a year later.

Double check to make sure you have the right bulbs in them,some bulbs may look exactly alike but one uses more amps then the other.If they are correct try cleaning out the sockets with elec. cleaner and then put some new bulb grease in the sockets.

If the bulb socket is melted you will want to replace the socket then use the socket grease (diaelectric) to keep the bulb socket clean of dirt and moisture. The diaelectric grease will help disipate the heat also.
If the bulb base was melted, you will want to clean the socket thuroughly and make sure no plastic has melted on the contacts causing a bad electrical contact.

If you have voltage in the socket with directionals then look in the socket and bend the tabs so that they will contact the bulb. They get hot and pull away from the bulb, rarely, but sometimes the tabs in the sockets will melt and break

the most likely cause of this problem is the hazerd flasher module, as it also works as the flasher for you signal lights

check to see if u have voltage pulse 12v at the socket with a test light . ir not you shoud repair wiring between flashr unit to signal socket. good luck

flasher relay(in fuse block)weak.also check for other burned out bulbs.

i had a bad socket that was my problem on mine.

check wire contacts in socket. If they are melted change the socket
It can be removed easily just unplug the wires.
I have replaced both front sockets on my 1999 Buick Century.
Hope this helps.
Ron