You do not have to send off your passport or BRP card when you apply - you just submit uploaded scans of them.
However, you CANNOT travel while the visa is being processed, because leaving the UK will invalidate your pending application.
Also, if your current visa expires before you get the new visa, your immigration status only remains valid while you REMAIN in the UK - if you were to leave, you will lose the right to live in the UK (and your application will also be invalidated), which means you won't be able to come back again without starting over from scratch by applying for a spousal visa from the US again, which would also reset your 5 years to ILR.
There is a chance that if you don't use the Super Priority service you MIGHT get your visa in time to travel in May, but I think that would be a massive risk as it can take 8 weeks for processing plus up to 2 weeks to receive the BRP after that (and processing may not start until after biometrics)... but counting from today, it's only 9 weeks and 6 days until May 16th.
So, I would definitely recommend using Super Priority.
