How far did they go with the visa transfer, and what is the situation with the current employer? You shouldn't worry too much about the new company, but instead

How far did they go with the visa transfer, and what is the situation with the current employer? You shouldn't worry too much about the new company, but instead worry about keeping your current job (and, by consequence, your H1B visa). Otherwise you may risk losing both, and have to return to your country of origin and start all over again.