Presumably, you and the new employer had the assistance of an attorney in filing the change-of-employer petition, and you should speak with this attorney for advice first as someone who is familiar with your case.
It is unclear whether this new-employer H-1B was approved as an extension of stay (with a new I-94 printed at the bottom of the I-797 Approval Notice).
If not , you are not in status now - there is no option but to depart and reenter using this new approval notice in order to be in valid H-1B status for the new employer. As long as the old H-1B had not expired, there isd no problem in leaving, and if the old visa stamp is still valid, it can still be used along with the new approval - you need not get a new H-1B stamp.
If it was approved with the extension of your status here, you can work right away without leaving - but you are correct that it is entirely possibly that the three-month gap may come up if you file for adjustment of status in the U.S. However, a provision of law called 245(k) excused lapses such as this if for less than 180 days. It appears that this might be applicable to your situation.
Speak to a lawyer to be certain.