I am a US citizen engaged to marry a girl with asylee status. What will we have to do before/after marrying to adjust for residence?

You need to clarify your use of the word "asylee". Asylee means someone who has been granted asylum. Do you mean your fiancee was granted asylum by either USCIS or an Immigration Judge? Or, do you mean that she has filed for asylum and is awaiting a decision? This greatly affects the advice an attorney can give on the matter. As for travel, if she is currently in proceedings before an immigration judge, she absolutely should NOT leave the US. If she was granted asylum, she also should not leave the US until she obtains a Refugee Travel Document or Advance Parole. You definitely should consult an experienced immigration practitioner in your area.