I’m a dual-National, can I apply for an H-1B visa stamp using a different passport than what is on my I-797A?

I hold two nationalities. I just got my H-1B petition approved. I plan to travel internationally in November and so I will need a new visa stamp.

Can I apply for an H-1B visa using my passport from one nationality that is different than the nationality I got my petition approved as?

If no, then can I switch my petition to the new nationality without going through the lottery again??