1.1) How to do characters like that right, people who are male on the outside and female on the inside and/or vice-versa?
I think you should mostly try showing that they feel comfortable and happy with their new bodies. Like... Well, it's someone who hated themselves their whole life, but now can have their biggest wish granted. Showing happy and comfort in this situation is probably the main thing to go with.
Do keep in mind, that it will feel a lot like a wish-fulfillment story IMO. The main thing I personally like seeing in Gender Bender is the internal struggle of the person, of them having to cope with their new body and the new societal expectations of them... But when you put someone who is a transgender into the gender bender scenario, there is no internal struggle, because you're giving them exactly what they want.
This is not a bad thing, but it will feel quite different from the more serious gender bender stories (I'm not saying this can't be a serious story, btw. I'm just comparing it to other serious gender benders... Since well, comedic gender benders are hardly comparable to serious gender benders) that actively deal with internal conflict of the genderbent character.
1.2) And people who were still dense of that?
Then I think you should try highlighting the confusion aspect and the internal struggles. Like... The person won't necessarily dislike their new body, but they might feel strange with it, they might feel like they miss something, or they might feel uncomfortable at first... There needs to be a process of acceptance, until the person understands themselves, until they realize what makes them happy about their current body, what makes them unhappy, and what they thought of themselves before the genderbent.
It's probably the kind of genderbender that I like the most. Where there is a lot of confusion within the character who doesn't quite know what they want, but has to keep on living and adapting to their new circumstance, trying to understand what is best for them, until they eventually start understanding themselves better and becoming happy being who they are.
2) What about people who were Cisgender male and/or female and then becomes of the opposite gender?
I think the main thing that should be highlighted here is the uncomfortableness with the new body? And perhaps a search to see if there is a way to get changed back?
Another way you can try doing it, is having the character become more and more apathetic about themselves. Plenty of trans people that have yet to realize they are trans, simply don't think about themselves. They may consider themselves as ugly or the like, but they don't actively worry about themselves, don't try to maintain their appearance and just... Live. Live while thinking of anything else other than themselves. Constantly occupying their head with work, school, games, books or whatever. In a very subconscious process of never having time to think of themselves, because they don't want to think, they don't want to acknowledge how much they hate themselves.
And yes, those are symptoms that can be found on people diagnosed with depression. They don't necessarily have to be depressive, but it's not strange for some depressive symptoms to appear on people that suffer from Gender Dysphoria.
And then there is also the option of actually writing about someone that doesn't necessarily fit into the female or male ends of the gender spectrum, because like... It is a spectrum, not a duality.
If someone doesn't fit that well with either side, they might be very confused about the change, like some things, dislike others, and be like... Unsure of what to do next. This is very similar to the "1.2" situation, but with the caveat that 1.2 is more about understanding that they are happy with the genderbending, while this one is more about understanding they aren't unhappy with the genderbending.
This could go in a lot of ways, like making the person accept the genderbending, or trying to refuse it, asking other people's opinions on which version of them they prefer (which is not the best of ideas, since it's something that they should decide themselves, as it is their happiness that is at stake... But hey, confused people relying on others isn't strange) and the like.