For H1B's you can apply for TPS but if you work for anyother employer besides one in I979 or I 129.. you will be terminated...Tht's my understanding... not much of help for H1B incumbent status but one can apply for TPS as a back up and not use the EAD.

But I have no idea how will TPS impact your travel. H1B you can travel without restriction but when you apply for TPS you need to get travel parole which is not 100% guaranteed.