Yes, a judgment can be transferred from one county to another, or even to another state without any notice prior to the transfer. It cannot, however, be made against a different entity. Given that you have the same name, I presume that it was a matter of leaving off a "Sr." or "Jr." designation. You should consult with an elder law attorney in your area.
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The creditor did not get a second judgment. Judgments entered in one county can be recorded in any other county in which the judgment debtor owns property. The problem here is that you have the same or similar name as your father; you showed up as having property on an asset search which the lawyer thought was your father.
Before you spend the money on your own lawyer, try calling the law firm and explain that the wrong party was identified and that you own property not your father. You might want to tell them your father is in hospice too. Anyway, see if they will voluntarily remove the judgment as it was against you. If they will not, then you will have to seek out, not an elder care lawyer, but a lawyer specializing in credit card debt. I would sue the law firm and ask for reimbursement of counsel fees for them being stupid and not doing their homework and discovering that the land is in your name.
This of course assumes that you are providing truthful information. Any lawyer you hire needs to confirm that this is indeed the case.