You are allowed to travel and even work in a different state so long as your primary place of residence (where you return to after work, pay rent, receive your bills, do most of your transactions etc.) remains in a state where your asylum application is pending. However, be aware that jurisdiction and venue may become an issue in a scenario where multiple states are involved and you may be required, by an immigration judge or the government attorney DHS, to show further proof in support of your current place of residence.

Please be advised that there are special considerations in each case and it is best to consult with an experienced immigration attorney.