^ It really depends on the trans person in question.
There ARE trans*people who identify as straight - but there are many who do not.
"Transsexual" describes people who have or have a desire to modify their bodies. To many this is not seen as a change, but as a correction of something they've felt was wrong all along.
"Transgender" is a more inclusive term. It can be anyone who has non-normative feelings about their gender. As you might suspect with such a broad term there are a LOT of different experiences.
Your cousin Jackie - does "she" still identify as female or does he identify as male ? Or something else entirely ?
There are many more terms and ways of describing things. "Genderqueer", "androgynous", et cetera. There are many ways of expressing gender.
I have known gay and straight trans* people. Some do see themselves very clearly as gay.
As a person who is part of the transgender community, I do feel strongly "gay". This is a personal feeling and doesn't project onto any other trans*people.
Not all trans*people date as if they were straight, definitely not. Say a person is FtM ("female to male"), I'd say the majority of these people would qualify dating a man as gay - if the trans* person truly identified as male, and not something else. I know many gay FtM and lesbian MtF - and of course straight ones, too.
A person who identifies as a transgender male may identify as gay or straight or queer or asexual or bisexual or whatever they do feel. Sexuality and Gender are not the same thing, this is clear.
However, they are good allies and have much in common and should not ignore each other - although I am all to familiar with the frustration of the T community when the LGB writes them off as a lesser minority.