A DC resistance of 10 ohms sounds rather believable. Offhand I would expect a lower resistance, based on a rather simple understanding of how it may be set up, but I don't have any specific values or very detailed design info to be be very certain. The DC resistance of transformer coils is by design quite low since any resistance there simply results in inefficiency and heating. The primary (and also the secondary, or at least a portion of it, since it's the same winding in an autotransformer) would presumably be connected more or less directly to the plug.
Signs of excessive heat at the plug prong is due to a poor connection right in that area, either where the wire attached to it, or between the prong and the spring fingers in the socket it was plugged into, or between the wire and the socket. Replacing the plug is a very good idea if you find it in that condition.