you have both the options
You have both the options, but remaining on H1/ H4 is considered to be safe option.
Hope you have several years for H1/ H4. Keep EAD/ AP/H visa updated. I have also read that it is good to carry AP when travelling overseas. I have not used it and have used it as well. In all the cases it has been a piece of cake. But never know, cake may not be fresh any more.