I haven't been not out and presenting as my target gender at the same time, meaning before I came out I presented as my birth gender, then made a total switch in my presentation upon coming out as trans, and then I was out to everyone. So my answer may not be applicable.
I still lived as female, and very femininely so, up until I almost turned 20. So I was like, 19 years and 11 months. Before coming out I started getting such attention rather late, when I was 17. I wished it had come a lot sooner and was eager to start dating and having sex as soon as that became available. I was very much in a hurry to grow up, cause I wanted to escape my horrible childhood, become independent and have my own autonomy. Adulthood wasn't just something I saw as an escape from my childhood, but was something that intrigued me. I was always mature for my age, and yearned for finding love and such adventures, at least since around age 8-10. But being bullied and very unpopular kept me from exploring that in my most of my teens.
After I started getting such attention though, still pre-transition, I was treated as a cis girl, cause I guess that's how I, probably quite obviously, appeared. And had a couple of "straight" relationships, with cis guys.
However, after coming out, I got a lot more "intense" (for lack of a better word), approaches from guys that were interested in me. It's only ever been other men showing sexual interest in me. Women have only seen me as a friend, or potentially cute enough to kiss at parties but never anything beyond that. I am gay but have been curious about/mildly into women before. With men it's been very mixed approaches throughout my transition. Some have pitied me, while others exotified me, while yet others have just seen me as a walking p*ssy and not cared about the rest it was attached to. But also a lot have been into me for a variety of reasons and just not been bothered by the trans aspect.
But then I'm also sure that my own perspective and approach to other men has shaped how they in turn approach me. I've always been very "out there", wild and open for all kinds of suggestions, especially sexual. Despite having dysphoria, I've always been very... suggestive, and confident in how I present myself. And I think that those kind of vibes I let off have a tendency to attract certain kinds of people who are perhaps a bit more direct and blunt with their advances. Cause, erhm, I guess I'm more or less known to be into that.
For me, I have definitely felt like somewhat of a target in the dating world, for better and worse, but perhaps not in the same way you described. Since coming out I've felt more vulnerable, exotified, judged, sexualised. But strangely that's also my card up my sleeve I can play to my advantage as well, cause there are definitely two sides to the coin of knowing I have something extra that attracts other people, but also objectifies me a lot, all things considered. I'm that weird kind of person who gets hit on by really weird, creepy guys but then also goes along with them all the way. Yeah, I get enjoyment from it, and a lot of strange stories to tell at parties. Most "creeps" seem to really just be socially awkward guys who didn't get much luck with their looks either, and not actually dangerous. A lot of them are really sweet, and surprisingly good in bed. So I give them a chance and I'm glad that I do.
So eh... how I'm approached since living as my target gender, is probably not exactly typical. But then neither is my approach to them. But I enjoy it, so I'm not complaining.