Where can I find information on how much an antiques White Rotary sewing machine is worth?

You start searching online. I assume that you have the model number and year, and have ascertained it's condition.
I have one from 1967, when I was 21. Not old enough to be worth anything, but I still use it and don't want to sell it.

sewing machines have little value as a collectibles, Some of the older treadle machines have small value if in showroom condition. You can list the serial number and a photo and we can help compare it with those being sold.

Check EBay or Craigslist.

There is 2 different numbers on the machine and I have no idea which one is the serial number so I will just give you both. 77-278382 is on a single silver tag on right bottom of machine and E-6354 is on a larger plate along with the manufacturer, no year, located on the back top.

Doesn't sound old or valuable. Mine has a much shorter number. Weighs a ton so I couldn't possibly ship it anyway, and selling is all about shipping. Put it on craigslist or on a sign at your local supermarket if you still think it might be worth something. And you still haven't indicated if it WORKS.