i am also having this same exact problem in my 97 taurus, after the car warms up to normal operating temp ( usually about half on the temp gauge) i turn the heater on with the fan dial set to the lowest speed and the other dial set to floor/vent mix. with the fan dial on 1 i can place my hand infront of the vent and i feel warm air, if i leave it on fan speed 1 the temp will stay warm for a few minutes.
however, if i turn the fan speed all the way up the warm air goes away within a minutes or so and turns to cold air ( not as cold as when i have the ac on but still colder then i want during the winter )
from my research on these forums heres what ive come up with
1. is the car overheating?
a.yes, look on your waterpump there is a little seep hole, if waterpump is bad there will be water coming out of that little hole.
b. if you have checked your water pump already you may want to replace your thermostat ( its like a 5$ fix )
2. ok so car isnt overheating lets move on to the heater core
a.with the car at normal operating temperature feel both of the heater core hoses they will be next to the fire wall towards the middle of the car if there both hot then you have other problems
b. if 1 is hot and the other is obviously cold then you know you have a blockage in the heater core and you need to flush.
c. heres where im stuck

both mine are hot well one is hot and the other is less hot.
so im going to try and do a radiator core flush then move onto the next thing.
3. assuming all these things arent the problem
a.replacement of the door actuator
from what ive heard these things can sometimes break and only allow a small amount of air in at a time.
how can you tell if its the door actuator that is the problem? well if youve already gone through all the above steps the only step left is the heater control module or w/e its called.