It could be a heater core. But I would not change it unless I was certain that it was leaking. (Disconnect both heater hoses and apply a 15 psi pressure test to the core itself.) I urge you to take another look at possible leaks in the engine compartment including a pressure test.
Here is a view of an older Series II motor on the passenger end showing where to look for leaks. Clockwise from far left:
Return heater hose fitting to tensioner assembly (o-ring)
Feed heater hose fitting to tensioner assembly (o-ring)
Tensioner assembly to plastic elbow (o-ring)
Plastic elbow to LIM (o-ring)
Water pump inlet to lower radiator hose (clamp)
Water pump weep hole (behind pulley)
Tensioner assembly to water pump (o-ring)*
*On newer L36 engines this connection is made with second plastic elbow sealed at each end with an o-ring.
With regards to the heater core, please keep these thoughts in mind.
1. When coolant level drops in the engine, the heater core is often starved of coolant. With no flow through the core, it quickly cools and will provide no heat to the cabin. No heat from the heater is, in fact, a good indicator of low coolant. Blown heater core will often put coolant on the passenger floor, rather than a puddle under the RF tire. Do you have coolant on the passenger floor?
2. Any moisture in the car during cold weather, including snow melt from your shoes, leaking door seals, or other water leaks can increase the humidity in the car to nearly 100 percent. When the glass is cool and you get in the car and add your moist breath to the already saturated air, it is common for the glass to fog.
3. If you have a coolant leak in the engine compartment, the smell of coolant is often drawn into the car through the heater ductwork. In my '95 for example, I occasionally get a good whiff of baked on motor oil caused by a leaking valve cover gasket that allows oil to seep onto my rear exhaust manifold. There is a rubber seal that is supposed to isolate the engine compartment from the heater'* air intake at the base of the windshield, but it is rarely 100% perfect. In a similar manner, snow and ice crystals can be drawn into the heater system where it melts and increases the likelihood for windows to fog.
4. A seeping water pump seal, or a cracked coolant elbow, that may only leak when the car is hot and there is some pressure built up in the cooling system, can produce the "couldn't find the leak over two weeks or months as the cooling system level drops" symptoms.
5. Most importantly, a large puddle of coolant under the RF tire points to an external leak in that area. Water pump, coolant elbow, radiator hose, heater hose.
So, while your symptoms may indicate a bad heater core, they may also be caused by these other easier to fix problems.
Did you fill the cooling system at the radiator with the engine cool before you looked for leaks? Adding to the overflow tank will not raise the level of coolant in an engine with a serious leak as you have described. Water is only good to test if the car will not be exposed to freezing temperatures.