I don't know the answer for current draw, but it should be somewhat less than the fuse rating. On the early systems there was indeed no switch and the thermostat button was the only control other than the passenger seat getting a weight switch attached to the seat springs wired in series with its thermostat.

It really does sound like you've got a short in the seat heater grid. The plastic heater grid simply has a resistive heating wire laying in cross channels as it winds its way up the grid. If a wire breaks or the wire turn between channels becomes unfastened, the wires in one channel may jump over into the next channel and short to its wires, effectively lowering the resistance, raising the current and thus raising the temperature.

You need to remove the grids for inspection. Here's a link to an old thread that includes pictures and a bit of discussion. Use the site search feature to find other discussions including tips on removal.
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Dave -940's, prev 740/240/140/120 Never overestimate what little I know.