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Parts for Whirlpool RF324PXEN1 Range - AppliancePartsPros.com

See the attachment for the tech sheet.

Sounds like a control board (Item 12 ibn section 02) problem.
The broil relay contacts have welded together or the electronics that control it are shot.
From the problem history I would guess it is welded contacts.

Looks like the board is obsolete.
That leaves you with a few options.

1. Hunt around locally and on the net for a replacement someone may have old stock.

2. There are companies out there that repair obsolete boards just Google "appliance timer repair"

3. Unplug the unit and the P2 plug at the control board. I am not sure if it is a separate plug or wires as I do not have a picture of the board. Then measure from P2-3 to P2-4 should be infinite ohms. IF 0 ohms then you know that the broil relay contacts are welded together.
The other possibility is to just gamble that my guess is right.
Then take the number off the broil relay and try to find a replacement. A company called Digikey has many of these. Also other electrical/electronic supply houses.
Then replace the relay on the board.
Best done if you have some experience or a friend with soldering experience.

4. And lastly replace the unit.