Jurisdiction for the lawsuit arises from (1) past contacts with the U.S. at the time of the claim from which the lawsuit arises or (2) current contacts with the U.S.
For example, if your LLC is being sued for...say, slander, that supposedly occurred 6 months ago, and your LLC were conducting business in the U.S. 6 months ago, your LLC would be subject to the jurisdiction of the U.S. courts based upon your past contacts. (Otherwise, businesses would just leave the country to avoid jurisdiction.) On the other hand, if you are currently in the U.S., but were not in the U.S. 6 months ago, you are still subject to the jurisdiction of the U.S. courts based on your current contact with the U.S.
Again, your personal assets would be protected. However, the LLC's assets would not be protected. Once you've been targeted by a lawsuit, there's no point in moving the LLC's assets overseas. If anything, you're more likely to have an enforceable court order freezing the LLC's assets in whatever jurisdiction you've moved to.
I assume that I'd still have subscribers to my service in the U.S., but it doesn't make sense to me that that would constitute doing business there. By that rationale wouldn't every internet company be subject to the jurisdiction of every country on Earth?
Accepting and continuing to do business with the U.S. subscribers constitutes doing business in the U.S. Under most double taxation (prevention) treaties, a software/online "service" is deemed performed where the customer uses the service, unless there is a significant human element to the service in which case the service is deemed performed where that particular human performs his tasks.