In a typical limited conservatorship the conservatee does not have any assets. My guess is that your petition for limited conservatorship would be successful unless your son has assets. If he has assets and the court requires a bond this could pose a problem for you depending on your credit rating as well as the criminal issues.
No attorney-client relationship shall be created by any responses to inquiries on this website. All of my responses shall be deemed to be advice on the general nature of the law. I cannot respond to specific cases without a complete interview with a prospective client.
As a general rule, although government agencies can access initial charges, what they consider are the actual convictions, unless there are some unusual circumstances. However, you actually should consult with an attorney who handles conservatorships, and who practices in the probate courts to get the most definitive answer to your concerns.