I agree with my colleagues. Your DWI probably do no affect your employer's H1B petition. If you have to apply for a H1B visa at a U.S. Consulate, you should do so after you successfully complete the program.
Filing the appeal will not extend your legal stay, and you will accrue unlawful presence. If your appeal is sustained, you will not accrue unlawful presence. You also have to stop working. Please hire an attorney and have him/her review the denial letter.