It belongs to all of the heirs of the estate-so each heir could consent to giving you a gift after receiving their share.
The answer given does not imply that an attorney-client relationship has been established and your best course of action is to have legal representation in this matter.
If it is estate property (not beneficiary designated to you only) then it goes in the estate and follows what the will says, period. If you aren't as well off as the others that is irrelevant. Perhaps the others will agree to let you keep it but that would be a settlement up to them.
This is not legal advice nor intended to create an attorney-client relationship. The information provided here is informational in nature only. This attorney may not be licensed in the jurisdiction which you have a question about so the answer could be only general in nature.
"I ... honestly just want to claim this unclaimed cash for myself." Why should only you get the money if the money belongs to other persons too?
This found money likely belongs to your father residuary estate. You should review your father's will to see who were supposed to get the residuary estate.
Getting money from the Unclaimed Property office is not going to be quick. The more money is involved the more proofs the office is going to require that the money is being released to the right persons.
"all of them are either completely financially stable or well off". Unless your father's will require that the persons is indigent to get money, whether they are well of is irrelevant to whether they are legally entitled to the found money.
You will need to satisfy the requirements of the Unclaimed Property office to get the money.
You can contact the person who handled your father's estate. Perhaps that person can get the money from the Unclaimed Property office and then figure out what to do with the money.