These are classic symptoms of a dead converter. The 120V AC side supplies the power to the converter and all the appliances that run on 120. The converter provides the 12V DC power. You are correct in that the lights and blower motor are 12V. Since the converter was not putting out any power, the battery eventually degraded enough that the lights dimmed and the blower quit.
I've had this happen, and replaced the converter twice in my unit with the exact same failure mode. Please note, that there are typically 1-2 fuses on the side of the converter itself which are there to protect you if you happen to accidently connect the battery cables to the wrong polarity. To trouble shoot them, disconnect your rig from shore power, disconnect the negative battery cable and pull out the converter. You can then access the fuses to make sure they are okay.
One final step I take to trouble shoot my converter is once it's out of the cabinet, CAREFULLY reconnect the AC shore power and measure the DC terminals to verify there is no 12V current. If there is still 12V, then the problem is most likely not the converter.