What happens to a mortgage when it goes through probate?

My dad recently died and I am doing the paperwork for summary administration in Florida (the estate value is less than $75K). One hitch in this process is that my dad owned a house in Texas that was mortgaged and has about $10K in equity. My sister currently lives in the house but is currently unemployed. My siblings and I have been paying the mortgage on this house while my sister lives there. My question is what happens to the mortgage on this house when the estate is put through the probate process? Is there any way for the house to be put in my sister's name without going through the finance process? I am afraid that my sister will not be able to get financing since she has no job right now. We don't want the house to be sold or for my sister to lose her place to live, so what legal mechanisms are available to have this estate go through probate and keep the house and keep my sister in it? Thanks.