This is a fairly common question. The issue is that in order to qualify for and collect unemployment insurance, you must represent that you are able and available for work. In order to qualify for SSDI, you must represent that you are unable to perform any work. At minimum, you're risking forfeiture and overpayment assesments, unless your state's unemployment statutes or case law specify otherwise. You should contact a local employment law attorney before you take any action.
You need to talk to a local attorney ASAP before you do anything further or make any phone calls.
DISCLAIMER: David J. McCormick is licensed to practice law in the State of Wisconsin and this answer is being provided for informational purposes only because the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship.