Does the blower run at all when you select heat? Fuse #37 is the fuse for the heater blower - it should be a 30 amp fuse (this is in the fuse panel under the instrument panel.) There is also a heater Blower Relay in the Power Distribution Box in the engine compartment. It is #K14 (or maybe R14.)
Another thing to check is the resistor block for the blower motor. It is located on the bottom of the blower housing, on the passenger side, inside the car, below the instrument panel. You will see a wire harness with 3 or 4 wires going to a plastic connector. If the plastic looks like it's melted, it probably is, and that may be your problem. There will be a screw holding the resistor block in place in the housing. If you can remove the plastic connector, do so, the remove the screw and the resistor block. You can get replacement connectors with pigtail wires attached on the internet. (Google "1998 Contour Blower Motor Resistor Connector" Personally, I like the kit from Dorman carried by several different auto parts suppliers. It gives you a new resistor block along with the new connector.) Check the connection pins on the resistor block. If they are dirty or have melted plastic on them, they need to be cleaned. Use a small wire brush or sandpaper to make sure the blades are "bright metal" clean, then coat them with a good electrical protection-type grease. (The little packets of grease auto parts stores sell for car light bulb sockets work well, or a good dielectric silicone grease is fine.) Once you cut off the old melted connector, solder the new connector pigtails to the ends of the old harness wires and cover the joints with heat shrink tubing.. DO NOT use crimp connectors! You'll wind up with the same problem in a very short time! Put everything back together and see if it works.
I've attached a picture of the resistor block and the connector. I found this at O'Reilly Auto Parts web site:
If you're not comfortable doing this kind of work yourself, find a friend who is, or take it to a good shop (usually a dealer is not what I would call a "good shop".)