you have three of the four already - so a volt meter or a 12v test lamp
would indicate them for you
you have a ground - the left turn/stop - the right turn/stop and the running lights

The best way to do the job is with an adapter that simply
plugs into the harbess below the tailgate and is already
wired correctly.  Takes about 2 minutes for complete job.
Almost any store that sells anything related to trailers,
boats or auto parts should have the adapter.  Then you won't
have a sliced and diced harness to cause further problems
later down the road.

Lugnut

The wire harness for a 4-pin connector is included with the connector.

WHT - Ground
BRN - Lights
GRN -- Right turn and stop
RED - Left turn and stop

You have a wiring problem in the trailer associated with improper grounding.
You described exactly the problem of a poor ground. The trailer's stop and
turn lights are getting a ground WHEN THE LIGHTS ARE ON through the running
light circuit in the truck. If the lights are turned off, then the brake and
turn indicators stop working too. (Actually, as I write that, it occurs to
me that symptoms should be exactly opposite -- the brake and turn indicators
work when the lights are off, then stop working when the lights are on.)

You have a grounding problem in the trailer or you have the wrong light
bulbs in the trailer. Odds favor a bad ground over the wrong bulbs. Your
bulbs should have two filiments in each, and the number stamped on the base
should be 1157.

On the trailer wiring harness (connector), WHT is ground. BRN is running
lights, GRN is right side turn and stop, YEL is left side turn and stop. A
common problem is that some guys bring the white wire into the trailer, but
do not select a suitable grounding method. You must put a good terminal on
the end of the wire, then use good star washers to poke through the paint
and press into the metal. (Paint is an insulator.) You should really scrape
the paint away from the grounding terminal to make a quality connection
point. Once you have your system working, then you can come back and paint
over the connection to preclude rust. It's a good idea to cover the
grounding point with tape before painting, then remove the tape to establish
the ground point. This is actually better than painting over a ground
connection.

Then, you have to ensure that the tail light housings are grounded well to
the trailer itself. This is a huge problem with trailer lights, they ground
through the mounting screws to the trailer itself, but the mounting points
are not qualified to provide the needed ground circuit. You might be
required to hardwire a ground from the housing (lamp base) to the trailer
frame. You _could_ bring the white wire from the connector all of the way to
the lights, but usually the white wire terminates on the tongue and the lamp
housings are then grounded to the frame of the trailer. If you wire this
way, then you will have GRN and BRN to the right side, and YEL and BRN to
the left side

CORRECTION TO AN EARLIER POST
I might have said that there is a RED wire to serve the stop and turn
indicator on the left side, this is an error, the actual wire color for that
circuit is YEL.

Trailer wiring is standardized, and a 4-pin wiring harness (by far the most
common) has WHT, BRN, GRN, & YEL. The arrangement of the wires in the tow
vehicle is important because because one might tow a different trailer some
day, and one would want any trailer to easily connect without having to
rewire something everytime the trailer changes.

Most trucks already have a 4-wire plug on them, whether or not they have a
trailer tow package on them. I don't recall that you mentioned which year
your F150 is, but I have a '95 Bronco that has a trailer connector, and my
previous '95 Bronco also had one -- they are both 6- or 7-pin designs, and
could be the result of a tow package. I've been shopping for F150s in the
past two or three years, and I noticed all of them on the lot that I was
looking at had trailer connectors on the truck.

When you buy a flat 4-pin connector from WalMart, or anyplace else, the
arrangement of the wires will always be the same, and any trailer will also
be the arranged the same way. The idea is that you should be able to hook up
any trailer and drive away, and the lights should work. Surely one would not
want to hook up to a trailer and have it start blowing fuses in the tow
vehicle, and this requires you to wire your connector the same way everybody
else wires them.