I assume you have a washer that says "dependable care" on the top, right?
Take the front off and look inside. 2 5/16 bolts on the bottom. It hinges out on the top, and will fall off at around 45 degrees. grab the transmission and try to spin it. Stuck?
Look under the tub where the transmission sticks up in there. Look ok, or full of muck/rust?
grab the motor pully underneath, stuck? If so, take the belts off and find out if the motor is actually stuck. If it is, it's a frozen motor.
See both belts there? do they look ok or screwed?
Leaking when filling, or leaking overall? There is a VERY common, very cheap, very easy fix if it is only leaking when filling. Unrelated to it being stuck, however.
What i'm trying to get at is there is a bearing/seal where the tranny sticks up through the tub. It can leak mud all over (flinging it all around inside at the level at the bottom of the tub) which is more or less death for that kind of washer. If it clamps down on the shaft, it won't move at all. The water pump is on a separate belt, however, so it should theoretically pump out, unless the belt from a stuck tranny is holding hte motor stopped. It usually wil just slip an burn off, however. Something in the pump can stick that and do a similar thing from the other side, but usually again the belts will just burn off.
A 1993 Maytag is a great washer, but if the seal is stuck, you will have a hard berking time getting the thing apart even if you have tools and experience doing so. That is the one thing (just about) that we won't fix on those. It's rare, but at over 20 years old anything can go wrong.
shaking violently is a sign that the thing the tub/tranny sits on is going out. It's cheap, but a P-I-T-A to change. You could also have a much simpler, free-er problem to fix- are the legs all the way down? I mean can you see under the washer? If the people who delivered it never locked the legs in place, they will slowly work down, raising the washer up on mini-stilts, and it will walk all over the place when spinning. Don't know how the shaking RAISES the washer, but it seems to happen on those for some reason. Also be sure to not overload and leave water level on MAX with those old ones, it will help balance the load while agitating and reduce shaking.
NON-HE alone is Speed Queen or buy something used. 
A speed queen should be expected to last as long as an old Maytag dependable care, but unfortunately they die in exactly the same way (seal/bearing). They are not all that different, which is to say they are great. Maytag even owned speedqueen for many years.
Don't get a cheap new toploader, the cheap ones die SUPER fast. Like, 4 to 5 years.
Water softener also means use less soap. Like, pods will ruin anything, use liquid and only use the bottom quarter of the cap or so. Too much soap ruins that bottom seal in old maytags, and will do so in a speed queen. On Samsungs, it does this:
The only washer i've seen that doesn't care about too much soap is the old direct drive whirlpools. If you can find a used 10 year old or more non-HE (non locking lid) whirlpool/roper/kenmore with model starting 106.xxxx(indicates that the kenmore is a rebadged whirlpool), you might get a good deal. Unfortunately at that age it will likely have problems, too. But fixable. Those washers are probably the #1 used ones i would recommend if you have a slim budget, simply because they don't have any 1 part that could fail and doom the whole machine, unless the specific timer or soemthing isn't available and that's what dies. We RARELY junk those, but it does happen.
I'd say bite the bullet and get a payment plan on a speedqueen. If you fix the old maytag that has been leaking and shaking violently for a while, it's just a matter of time. Lets say the motor is frozen and you can just change that. Good news. Now why is it leaking? http://www.appliancepartspros.com/07-tub-parts-for-maytag-lat7793aal.html check out part #27, that's the common leaker if it's when it's filling. Simple and cheap. If that plus a motor is the problem, great! Motors are pricey, but that little piece is cheap. Now, why is it shaking around? Could be the legs, could be a couple other htings. Legs are free to just screw back up as high as possible (while leveling the washer) and lock them. You see it's kindof a slippery slope with old machines. You could fix all of these problems and that tub bearing/seal locks up next week. then you're berked, and you have an expensive motor you just bought and a lot of time put into the washer. Even if you fix it completely, that age of machine is just a matter of time. Start saving for a speed queen.
If you open it up and you see it as clean as this:
Then try to fix it. Some of them we run into are literally as clean as new inside. If you take the front of and inside you find a horizontal line of mud and muck slung around about halfway up, your machine is dead or dying.
Not a dependable care, oddly. 
