The sound you hear is magnetic hum of the water valve solenoids. The machine's pump and drive motor have nothing to do with filling water and do not run until the target water level is reached. Water flow into the machine is by household mains pressure.

There should be screens at the faucet-ends of the supply hoses if your installation was done correctly. Check those first.

For sure there are screens inside the water valve where the hot and cold hoses connect. They may be plastic or metal. Either type can be carefully removed for cleaning. Avoid distorting or tearing metal screens, debris can pass through.

I haven't worked on a ThinTwin set so can't give you specifics on what to disassemble to get at the valve. Observe where the two water supply hoses connect to the back and proceed accordingly. There may be videos on YouTube.

Your machine is 16 years old. The serial number indicates manufacture in the 27th week of 1998.

Thanks so much! I will look into this.
Very grateful for the advice.

Regardless, now here is my question: I have it out and cleaned it. A lot of debris came out, mostly iron and manganese, etc. I put my mouth on the end piece and blew, hoping to get water to go through but hard as I blow, I can't. I'm wondering if that is b/c the solenoid is not functioning (since it is disconnected)? If that is the case, I have probably fixed the cloggage/problem and can re-install and back in business? Otherwise, I don't mind buying a $16/ part.
Next question is couldn't I likely find a local store that sells it so I don't have to wait for it to ship?

This one on Amazon could be yours by Tuesday:

http://www.amazon.com/gp/product/B0024IAQ9G/ref=oh_aui_detailpage_o06_s00?ie=UTF8&psc=1

It works on almost anything Maytag/Whirlpool.

Correct, you can't blow through the valve when the solenoid(s) are not energized ... same as water can't flow through.

It should have much better flow now ... those metal screens can be pried out for better cleaning but as I said above be very careful not to tear the mesh, and reshape them as closely as possible to the original form for setting back into place so debris in the water supply can't bypass.

Replacing the valve is a reasonable alternative if you want maximum flow and don't want to bother with more in-depth cleaning.

Regardless, now here is my question: I have it out and cleaned it. A lot of debris came out, mostly iron and manganese, etc. I put my mouth on the end piece and blew, hoping to get water to go through but hard as I blow, I can't. I'm wondering if that is b/c the solenoid is not functioning (since it is disconnected)? If that is the case, I have probably fixed the cloggage/problem and can re-install and back in business? Otherwise, I don't mind buying a $16/ part.
Next question is couldn't I likely find a local store that sells it so I don't have to wait for it to ship?

sorry for double post...don't know how that happened.

I just cleaned the screen and flushed the hoses as you recommended and it is back like a champ! No new part needed. I've never seen the tub fill up so fast as it is now. Man, its been clogged for YEARS i reckon. I bent the smack out of the cover getting the valve out, had to jury rig a comeback, back in action!