Hi @Phillip Garcia. I'm no lawyer, but I think you need to keep a lawyer involved.
However, I'll share what I've heard others do.
First, keep in mind that, yes, they are going to want some money. Frankly, that's only natural. Maybe $10k each will "buy them off", but I don't think so. Why? Simple: Beach House. Washington State.
I don't know the area that is in, but it is most likely worth a goodly sum, especially if it is free and clear.
I would recommend figuring out how much the home is worth (I'd be happy to work up an CMA for you) and dividing that and the $30,000 by the 4 heirs. Then, with an agreement in place, your mom can get a home equity loan on the beach house and use it to pay off your cousins.
Maybe you don't do it in even 1/4 size shares, but they are going to expect some sort of value from the home.
You could also look at giving them rights to come out to the home for one month out of the year - a free vacation home if you will. They might need to be on title for that, or it could just be an agreement. But thats why it would be useful having a lawyer involved. If she no longer wishes to use the probate lawyer, I can recommend someone who specializes in real estate law.
As I said, those are just ideas, and I am not a lawyer. Get professional advice before you do anything.
Send me a message and we can chat more.