** Kenmore Washer model 110.20842990 filled but would not agitate. **
What did the timer do at that time, progress to agitation and past or stick at the fill portion of the cycle and never move further?
** I thought that the fill switch was the same as the fill valve? **
"Fill switch" is very vague terminology.
It is the timer that initiates a fill by closing an internal switch contact, the water pressure switch (aka 'water level control') that stops the fill at the appropriate height and the water *valve* (at the back of the machine where the water fill hoses attach to) that opens or closes when electrically powered to allow or stop water from flowing into the machine.
** The fill switch is what I tested and it had shown good continuity on the multimeter. **
If by "fill switch" you're referring to the water level control, it is a SPDT switch with both normally open and normally closed contacts (see the following link). Which set of contacts "had shown good continuity"?
* It now will not stop filling unless the machine is unplugged? **
Will it overflow if left filling or will it never reach the desired level?
If it will overflow, it sounds like the water level control is either not properly sensing the water level (maybe due to a problem in its air supply tubing) or it is not acting upon that information once received (ie. the switch is defective).