tell your new boss

to apply H1 for you as soon as possible since your J-1 already got waivered and DOS will object your extension of J1 either from new boss or old boss. However, you should be legal before the expire date on the form 2019 even if you might not get pay from your old boss for the last month(you could tell your boss that the form is a legal document for your staying and you could sponsor yourself if his money run out since it's only one month. In addition, you will also be fine if you get your receipt of H1 before the end of July or within 2 month's grace period. Good luck!