if you have copies of paid income taxes past year bring it down with your new work work permit with new company. No harm exhausting every options you may have.
As Tod mentioned above, with a valid work permit in TH you should be paying income tax and bring that last 3 months income tax paid form 19Bis when doing your extension of stay based on marriage.
What Robert Lagas just posted below for online application is good. You should apply online and wait for their reply. What I gather from couple of friends here, they got their non b w/o showing their degree from SG. But current company documents must be completely thorough.
nope only singaporeans for now as SG is considering giving the fast lane to Thais as well. And well they claim Singaporeans frequent bangkok on weekly basis etc.