What would happen if I don't present myself to do jail time because I need to leave the state for an emergency?

If you have been sentenced to jail and given a date to report, and you fail to show up on that date, a warrant will issue for your arrest. You really should consider other options. One possibility is having your attorney file a motion to postpone your reporting date in which he or she explains the emergency. The Court may give you some extra time so that you can take care of your emergency. If you don't have counsel, but are on pre-trial release, you might talk to to your pre-trial release officer and see if he or she can assist you in requesting a delay. If that fails, you might try contacting the court clerk and explaining the situation, and then ask if there is any way you can get the matter set before the judge prior to the start of your sentence so you can ask for a short delay to take care of the emergency. Again, ignoring the reporting date is the worst choice you can make.