The President's policy announcement should not interfere with somone's eligibility and/or ability to apply for permanent residence. That being said, if you qualify for residence, I would suggest you take that route.
Once the rules are announced, which should be by mid-August, things should be clearer. You and your boyfriend may wish to consult with an experienced immigration attorney for further professional assistance and guidance.
You should have received a full transcript of all the hearings before the IJ. No transcript is prepared until and unless an appeal is taken. The last IJ hearing, at which the IJ granted asylum, is probably the first or last of the transcripts, depending on the arrangement of the copies. The most you can do is request an extension of time from the BIA to respond to the government's brief.
Of course, retaining an experienced immigration attorney may be a tremedous help to you as well.
If you wish to change the name on your card now, you will have to file Form I-90 and pay the required fee of $450. It is probably easier (and cheaper) for you to wait until you apply to remove the conditions and indicate that you are taking your husband's name then.
As I wrote in answering your prior posting, you need to consult with both a criminal law attorney and an immigration attorney as soon as possible. These arrests, if not tended to properly, can be major problems for you if you end up being convicted on the prostitution charges. The criminal attorney, working with the assisatnce of the immigration attorney, may be able to get the charges reduced to something which will not pose a danger to you.
Don't waste time; you need to attend to these...
I-601 waivers can be challenging. I have found psychological evaluations of the qualifying family members to be of value as part of the evidence presented. I would also say that you would be best served by retaining an experienced immigration attorney to help.
You will be able to adjust after the refiling if you are in legal status. Filing an appeal or motion to reopen DOES NOT give you lawful status.
If your employer retained an attorney for this, speak with that attorney.
No attorney, whether practicing immigration law, personal injury law, or criminal law, will EVER counsel someone to lie on any application, whether it is to the US government or a foreign government. It should therefore be evident to you that no attorney, whether on AVVO, some other website, or in person, will even suggest you should lie.
Aside from that, it is highly unlikely that you would be able to cover your tracks as well as you think, and once your lies are revealed, you wll have more...