Ok, so I do have a blower motor resistor. From all the online searching I have been doing, it seems that I have the manual controls but I have a blower motor resistor for the auto controls. This type of blower motor resistor is also called a "blower motor fan control amplifier".
This is what I have.
The cheapest I have found it online is 83 bucks plus shipping.
I also found this
tutorial on how to repair it, but I have no idea what parts to buy to repair it. The links in that write up say the item is obsolete. I am very good at soldering and I am quite sure I can repair this if I knew what parts to get.