This looks like a typical Kenmore/Whirlpool model. The parts are relatively standard and readily available through Amazon (for the most common parts) and both online and local appliance part stores.
The water problem is likely the inlet valve. I replaced that last year on my Kenmore. Part # looks like it should be the same: 285805 Here is an example from Amazon:
Apparently having problems more with cold than hot is quite common - I had the same situation. This part is pretty easy to replace.
The other problem is likely the clutch, though it could be a bunch of other things. I have replaced the clutch, motor couplings (which cause some other problems when they wear out, and those are actually designed to wear out rather than kill the motor which would be far worse), agitator dogs and other parts several times. Most recently I got this kit from Amazon:
which again seems to be (according to a search on partselect.com) the same parts needed for your model. I would get a full kit, because it doesn't cost much and odds are that if the clutch is going (which I think is your main problem) that the other parts are not far behind.
Unfortunately, the clutch replacement and related parts (except agitator dogs - those are trivial to replace) is much harder than the valve. It basically involves:
- Disconnect power (unplug) and water supply and drain hoses
- Remove the agitator (usually a few plastic parts and one or possibly two big bolts)
- Tilt the machine onto its back
- Remove the pump (two clips)
- Remove the motor (two clips with screws)
- Remove the gearcase (big heavy, three bolts)
- A bunch of small parts, including some really annoying (easy to remove, hard to reinstall) clips
YouTube is your friend. Watch a couple of videos (look for "replace whirlpool washer clutch" or similar) before you get started.