Pixel support is dead, what ROM to flash?

Last time I did any ROM flashing was on my Galaxy S4 after Samsung stopped the updates.
Looks like LineageOS does do over-the-air updates if I read the site correctly, better then old Cyanogen which needed new ROM to be flashed, and any problem was always solved by format phone and reflash.

Any other Pixel ROMs that might work better on the Pixel?

I did see that Google will offer 1 last update in December, but may not wait that long.

Which specific model of phone is it? The original Pixel?

Yes, Pixel 1. Google announced that December will be the last security update, did not get one this month.