I would question the credibility and experience of your new employer. It it a consulting firm? Is it a full time job at a midsized or larger employer?
Your family's H4s are related to your H1B and not your employer. There is no "transfer" of employer needed for them. Your family needs only the approval notice of your employer at their time of appearance at the border and valid (unexpired) H4 visas which they apparently have. If you have a transfer in progress and do not have an approval notice for that employer as yet, they need to delay travel. You can pay for priority/premium processing in order to get a fast decision.

