Washing machine water flows out of bottom as it fills and while it washes

The odds are that this means a crack or tear in your rubber boot that seals the bottom of the tank.  This is generally not a DIY repair and the cost of the replacement part can be way more than buying a similar (working) older unit at a garage sale.

It is also possible that some drain hose (or supply hose after the water valve) is dripping and you could find this out by removing the back sheetmetal panel and start filling... watch for where the drip comes from.  If it is one of the hoses, this would be much easier to try to fix.  With the back open you will be able to see all the  possible culprits :)

As a caution, with the back panel removed, the wiring is exposed and could be dangerous if you dont take the necessary precautions.

One other test you could do with out opening the back would be to slip a piece of cardboard under the washer after it has been sitting for a while and stopped leaking. Let it sit for a few minutes to ensure there is no leak (slide it out to confirm no leak).  then pour in about a gallon of water carefully into the tank  and see if the leak starts up again.  If so, this pretty well indicates the boot is leaking and you need to 'give your washer the boot'  :)

If it is not leaking , then pour in some more and if still not leaking,  think about opening up the back and finding where the leak is.

Hope this helps!