I have an 86 Ford F-150. For some reason the engine still tries to run after I turn it off with the key. Could this be a carburator problem or ignition problem?

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Lhydro101 - Just how long does the engine "run On" . - It Could be an ignition switch prob. - sounds like you did not turn off - simple - replace switch.
It could be your engine is "dieseling" - too much carbon build up that gets hot & keeps fireing. - Runs rough & lumpy - more serious -needs major work.

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98 F150 S/C 4.6 2wd Vermillion/Silver,HD Tow Pkg, Boards, Vents, Bug Chaser, DIY "breezer" Tailgate, 5th whl hitch, Tool box, 16" OEM Mag's,
Mobil1 /Wix Oil filter/ K&N / air box mod.
In looks & go - Chevy - like a rock.

 

If it is dieseling, somtimes replacing the spark plugs will help, as you can get carbon deposits on the plug. Essentially, the hot carbon keeps igniting even with the ignition off. On a carberated engine, you can suck in enough fuel vapor for it to go on for awhile. Pretty embarassing when you pick up a girl on a first date!

If the carbon deposits are on the head or pistons, changing the plugs won't make any difference. If you have a bad valve, it can also get heated enough to cause the deiseling. Cross your fingers and hope a new set of plugs buys you some time.