I am somewhat sympathetic to the idea that if you need active filtering not to say awful things then you should fix that or be socially punished for it. Think "white people trying very hard and very obviously not to look racist".

I don't think you said anything particularly horrible. You are clearly underinformed about transsexuality (e.g. your equal oppression assumption), and the less-than-nice things you said most likely stem from that. I will now alienate the social justice blogosphere by saying that you do not have a duty to drop everything else until you are adequately informed. I do believe that you should learn the 30-second version:

• "Tranny" is, as Alicorn said, a (somewhat mild) slur. I'm not sure how much filtering it requires to stop using it, but I'd be surprised it it were an unreasonable demand.
• Other slurs include "shemale" (extreme slur), "hermaphrodite", and any pronoun or combination of pronouns other that those used by the person.
• For your purposes, a trans woman is and has always been completely a woman from the moment of conception, and her life as a boy was due to parental error (ditto for men and non-binaries). Failure to completely alieve that is a faulty intuition on your part.
• There are a lot of stereotypes about transpeople. You cannot be reasonably expected to never propagate them. It is considered good form to say "Whoops, sorry" when they're pointed out to you.

Hey, is that a Japanese cross-sex-reincarnation compersion song? Awesome.