Your Model Number looks incorrect the following are for a FBD not a RDB.

Here are your parts
Replacement parts for WHIRLPOOL RBD305PDT14 | AppliancePartsPros.com

Here is a tech sheet and a service manual
http://www.servicematters.com/d...0Sheet%20-%208302068.pdf
https://secured.whirlpoo...524e76/$FILE/8178007.pdf

Sounds like a keypad problem but could be the control board. Unfortunately I cannot find a pin-out list for the keypad so that you could check it with a meter then if it checks out the control board becomes the prime suspect.
If the below cleaning does not work I would replace the keypad first but it is a bit of a crap shoot or call in a service tech.

Connector Cleaning
Unplug the unit
Open it up
Remove the ribbon cable from the keypad to the control board at the control board.
Clean the edge connector.
If it is the type where there is not a connector just the insulation is removed, it can be cleaned with a pencil eraser. Do not use an ink eraser as it is too abrasive.
Check the tracks in the ribbon cable for hairline cracks. A magnifying glass helps here.
Would not hurt to clean the connector on the board. Use electronics contact cleaner or isopropyl alcohol and a tooth brush. Be gentle and do not reconnect till all the solvent evaporates.
Reconnect everything and give it a try.

The oven preheated and stayed on for about 20 minutes and then went off.
When you say went off, what exactly do you mean re: did just the elements turn off or did the whole unit shut down or something else?
Did the unit come back up or does it still not work?

Do you think we need to purchase both the control panel and the control board?
Cannot really say without answers to the above.
The control panel would be the prime suspect as the keypad should not be able to shut the unit off.