Do you feel any heat coming out? My interpretation of your description is that when you put the T/S on fan, the fan will run without any heat. If so, then this would be what I expect. The "Fan" setting just runs the fan continuously. Not much help in winter.
The best thing to do (if you are the handy type) is to run the built-in diagnostics on the furnace. There should be a full description on the inside of the front panel. Basically, it will exercise each major component in the furnace, and you can see what is working and what is not.
Here is a snippet of the test procedure from the manual (note that yours may be slightly different from this):
Component test—The furnace control board allows all components, except gas valve, to be run for a short period of time. This feature helps diagnose a system problem in case of a component failure. To initiate component test procedure, short (jumper) the TEST 3/16-in. quick connect terminal on control (behind the Y/Y2 terminal) and the COM-24V terminal on furnace thermostat connection block for approximately 2 sec.
NOTE: Component test feature will not operate if any thermostat signal is present at control board. [WH: unhook red thermostat wire during this test]
Component test sequence is as follows.
a. Momentarily jumper TEST and COM-24V terminals until LED goes off. [WH: Use a new wire from your junkbox]
b. LED will display previous status code 4 times.
c. Inducer motor starts and continues to run for entire component test. [WH: You'll see it run]
d. Hot surface igniter is energized for 15 sec, then deenergized. [WH: It will glow red-hot]
e. Blower motor operates on FAN speed for 10 sec, then stops. [WH: You'll hear it run]
f. Blower motor operates on HEAT speed for 10 sec, then stops. [WH: You'll hear it run, maybe slower]
g. Blower motor operates on COOL speed for 10 sec, then stops [WH: You'll hear it run, maybe faster]
h. Inducer motor stops
Please note, if you are not handy, find someone who is to do this!
Hoping you have heat soon,
Wally H.