You sound like a very nice and caring person who really wants to do the right thing.
Remember that you are making a gift and anything you do is not legally required and hopefully the children will appreciate it and not look a gift horse in the mouth....
I would try to outline the value at the time of death (I am assuming it was in his name only and worth $13,000, net of any mortgage, and assuming under your state's law you would inherit 50% of the value, making the value of their "gift" in signing it over $6500).
I would also tell them what you put into the house of your own money since to pay off the mortgage ( willassume there was none) and to fix it up with improvements (say $1000) and the money you paid to fix it up for sale (say $2000) and the sale expenses you paid and the costs of sale such as broker's commission, legal fees, deeds, transfer tax, etc. (say $1500), which is a total out of your pocket of $4,500. Subtract that from the $20,000, getting $15,500. You might either split the $13,000 or the $15,500 50-50 with the kids as a group. If the latter you'd give the 4 kids 1/8th of the $15,500. But what ever you do, you will be doing something most people would not expect. And doing yourself and your late husband's memory proud.
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