well the heat keeps coming out because the heater control valve isn't closing.
you can test vacuum motors by themselves using a vacuum hand pump to see if they are working correctly.
but you need to chase down why you are not getting vacuum to the motors.
usually the climate control arm that controls the vacuum signal has got crud inside it and that had stopped up the vacuum getting to the motor. you will need to pull it out of the car to work on it, so it might be a good idea to test the motors first just to make sure they are working. that way you know the problem is inside the control block in the climate control manifold.
there is a vacuum plug junction that we call the octopus this is connected to the rear of the climate control manifold.
basically its a large block of rubber with all the vacuum lines running out the rear to feed all the vacuum motors behind the dash.
when you service the control manifold you will want to inspect these rubber hose manifold to make sure nothing is cracked from age and leaking vacuum signal. it can be a source of idle vacuum leaks and running problems.
once the climate control manifold is removed from the lower dash you can inspect it and try to clean it out using compressed air and electrical cleaner, a pipe cleaner would not hurt. each vacuum tube should be clear so when you move the selector arm to different points you can blow through a hose connection and have to come out the black vacuum control line which is the master vacuum feed signal. if you find you cannot blow through one of the veins then you have most likely found the source of the clog.
in your case it appears that 2 lines are clogged.
i would also take the time to inspect your vacuum reserve canister in the engine bay on the passenger side shock tower.
attach a tube and blow through the inlet on the vacuum can then try and suck air out,, if working correctly the air will only flow one way. if you can move air into and out of the vacuum canister then the reed valve inside is blown, replace the canister with a new one.
one source of clogged up A/C vacuum systems is carb backfires all you need is one and it can blow the reed valves and send junk into the climate control system, it can also damage the vacuum motors.
now usually the heater control valve dies from the coolant and heat eating it, so again i would test it first. you can close it manually but i would try hooking up a hose and then either use a vacuum hand pump or your mouth to create a vacuum and see if the valve moves.
if later on the valve works but the heat will still not turn off, that means the little valve inside the center of the heater control valve has rotted away, so i would just replace it.
testing each vacuum motor can be done with the dash in place you just need a long hose to reach them and long arms and fingers.
most likely you have a clogged up climate control vacuum manifold causing the problem Worst case you will need to replace it from a donor car and make sure it works before install.
I had to replace my original as nothing i did would get the clog out.
any who you have some diagnostic work to do this weekend, and again if the vacuum motors are all working and not leaking then start thinking about pulling the octopus and taking the center dash apart to get the climate controls out of the car and on the bench for cleaning or replacement.