If you get married to a US citizen, she may indeed file a relative petition on your behalf. Depending on how fast the DHS sends your case to the immigration court, you may be able to file your adjustment application with the CIS. If your file is quickly sent to the CIS, you may find that a notice to appear has already been filed before the date of your marriage. In this case, you may apply to adjust status before the immigration court.
As you have some immigration history, the DHS will likely give extra scrutiny to your case. You must be sure to properly document your relationship with your spouse and your intention to make a life together as a married couple.
You may remain in the US while waiting for your green card to be processed. No need to worry about the asylum clock any more, your asylum application has already been denied. Unless your file gets misplaced by the CIS, your next stop will be the immigration court. You may seek all available relief at the court.
A marriage to a US citizen will not cancel an asylum application.