Same sex married couple for dependent's visa

The information you found is only partially correct:
There are no same-sex marriages in Singapore (and, as you probably know, the issue is socially sensitive, as large parts of the population reject the idea of same sex relationships altogether). Sex between two males is even illegal due to an old, colonial time law (but curiously, between two females it isn't).
On the other hand Singapore is, by international treaty, obliged to recognise foreign marriages. That includes same sex marriages from countries where they are legally possible.
There have  been a (very small) number of family reunion visa / DP issued in such cases. But obviously the authorities dislike this fact and (as you have encountered) will do anything they can to discourage or delay your application (short of saying it isn't possible, which they legally can't).
You might be lucky in the end if your partner is a type that is very welcome here (highly skilled in a niche with severe manpower shortage, world-known scientist, artist or athlete, etc.) and if the authorities are covinced that you will keep a low profile and not make your family situation widely known.
If you want to fight for your rights here (or gay's rights in general), you will lose.
I recommend you apply for a different kind of visa (e.g. for work), if you have the opportunity for it.

I know a Singaporean-German gay couple who, due to the known difficulties, decided to marry and live in Germany instead. They are happy there.

There also was, a number of years ago, a public discussion about whether same sex relationships should be de-criminalized. At the time the authorities concluded that this would not represent the will of the majority and thus should not be implemented. I think this observation is probably correct - the Singapore society isn't ready for it!
(Please don't blame me for writing this - I do have a different opinion about gays and am just trying to state the facts here.)

beppi wrote:

I know a Singaporean-German gay couple who, due to the known difficulties, decided to marry and live in Germany instead. They are happy there.

There also was, a number of years ago, a public discussion about whether same sex relationships should be de-criminalized. At the time the authorities concluded that this would not represent the will of the majority and thus should not be implemented. I think this observation is probably correct - the Singapore society isn't ready for it!
(Please don't blame me for writing this - I do have a different opinion about gays and am just trying to state the facts here.)

Thank you so much for your insights. I think that our best chances is indeed applying for separate jobs - which we have. As mentioned, he is already in Singapore and my application is currently pending with 3-4 months waiting as per MOM. I now wonder if my inputting in data that I'm married to a man has played a role in this unprecedented delay in my visa status...