Keep in mind that 'winning' the lottery is no guarantee that you will 'win' a greencard ... they draw more names than the number of visas available.
DOS is correct CIS will determine your eligibility to adjust.
To find out early, take all your asylum and/or court papers to a meeting with an immigration attorney ...each case is different ... thus, this blog probably won't be able to give you the 'secure' answer you seek.
firstname.lastname@example.org -- 503-803-0055
FORMER IMMIGRATION LAW PROFESSOR -- LEGAL DISCLAIMER: This answer is offered for informational purposes only. It does not constitute an attorney-client relationship.