They will likely show up on any background check--even one run on an apartment in Dallas. You'll need to contact attorneys who practice in the counties in Indiana and Kentucky where the cases were filed. They'll be able to tell you your options based on the laws in those states and the dispositions of the cases.
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Each state has its own laws and lawyers are licensed by each state. So, you really need to address your questions to lawyers in Indiana and Kentucky. You might want to re-ask your question on avvo and where it asks what state you want to know about enter each state. That probably means rea-asking twice.
Answers on Avvo are for general information purposes only and should not be relied upon as legal advice. No attorney / client relationship is created by providing this answer. For specific advice about your situation, you should consult a competent attorney of your choosing.
Out of state felony convictions will usually appear on a basic criminal history search.
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