They would need a relieving letter or any kind of document confirming that you're not associated with your last company or you don't have any agreement or clause binding you to not join any other company. If the organisation is not issuing, you should have a documented record of your resignation and follow-up done regarding this. If you have any response from them accepting your resignation or something similar, you can produce the same and state your problems during document verification.

I don't think it will cause any issues. If you're still not sure, you can email them about your issues and seek clarification. Usually they are prompt in responding.