Yes, your criminal history should not affect your ability to legalize your husband. Only your husband's criminal history matters. I suggest that you contact an experienced immigration/criminal defense attorney for a face-to-face consultation and give him/her all of the facts surrounding your situation. He/she would then be in a better position to analyze your case, advise you of your options and, if possible, make sure that the necessary paperwork and documentation is presented so as not to delay the process.
Whether you can still fix your husband's immigration papers will depend on the nature of the felony conviction Both you and your husband should consult with an immigration lawyer before you do anything.
If you are a U.S. citizen and want to sponsor your husband for a green card, your criminal history does not matter. Whether he can get a green card depends on other circumstances (did he enter the U.S. legally, how long has he been living here, does he have any convictions, was he ever deported, etc.).
All immigration questions are complex, fact-intensive and cannot be answered simply. The short answer is "maybe" and "it depends." You need to speak with a qualified immigration attorney for this issue.
As the American, it does not matter if you went to jail or prison with a couple of exceptions. If you are the alien, a felony may make it more difficult or not possible depending on the crime(s).
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