Air conditioner starts .but after short run engine heats up and air goes hot .1999 lincoln

The cooling fan isn't causing the engine to run hot, it's electrical and cannot heat an engine. It is turning on in response to your engine running hot. Check coolant level and age, if it's still running hot then check the thermostat - it could be stuck closed. If the thermostat is fine and coolant level is full, it could be partially clogged radiator or head gasket problem (a sure sign of bad head gasket is that your coolant keeps disappearing with no leaks).

it has own head gasket on #6

If the engine is not getting up to normal temp, I would replace the thermostat. Both heater hoses have to be hot to get heat inside the car.

To start with, you are talking about the temp gauge. The thermostat is in the engine.
The coolant system could be low or have air pockets and still look full.
First thing I would check is the cooling fan, then the coolant level.

Air pockets will make the guage do that, and also cause your heat to come and go. Also, a poor radiator, as in rotting, or plugged up, will cause high temperature issues.

Since your blower is working and it is a heat issue,thats where we'll start,First thing is have you checked the coolant level in the radiator when engine is cold,make sure it is filled to top,next verify you have coolant in reserve ,Start your van and get it up to normal operation temperature,If your heat gauge on the dash works correctly it should be 190 degrees,Now lift the hood and locate the two heater hoses going into the heater assembly at the fire wall,Both hoses should be hot to hold on to,if both are hot and no heat inside your problem will be the blend air door in heater assembly.
if neither hose is that hot you need to replace the thermostat in the engine to get it up to 195 degrees.
if one hose is hot and one is luke warm then the heater core is blocked and to blame. hope this helps you.

Chk for a vacuum controlled heater valve in the system. Usually on a hose. When the engine is not running it is open, but as soon as you start the engine, it closes and soon after the coolant in the core is stationary and cools down. GoodLuck _Ned_

Get your ac filter replaced. My car did the same thing for a year until I figured out what the problem was.

No, bad thermostat.

if the heater core is getting hot then the problem is the air temp control blend door actuator, a common problem.