This question is extremely complicated, and the right answer is "contact an attorney experienced with refugee/asylum applications before you do anything".
Applying for a J visa requires a letter from the ministry of health from your home country, which could be a problem. It appears you may be able to apply for asylum while on a J visa. Refugee status and asylum status are two different things with two different sets of rules / processes. And then there's a J1 waiver based on persecution, which is yet another thing. Once you have refugee or asylum status, then you'll have an EAD and won't need a visa at all.
Very complicated. Don't try to do this yourself - that's you you end up screwing it up and getting deported.