Sleep however you normally would in whatever makes you comfortable, if you wear pyjamas then keep doing so, if you sleep naked or in underpants then just sleep in a diaper etc. The more normal for you that you make it then the more normal and natural the nappy will feel, helping you sleep better.
As for being kept awake by the nappy/knowledge you're wearing it, try putting it on a little before you get into bed (an hour or so maybe) so your body adjusts to feeling it before you lie down to sleep, also simply wearing them more often will help your mind adjust.
This doesn't mean just put them on more often right before you pee, it means put them on when you really don't need one yet and then just wear it like regular pants until you do use it, this will help wearing for longer periods and will "naturalise" the wearing of them so you mind isn't constantly thinking "I'm wearing a nappy, oh boy I'm wearing a nappy, this is great, I'm going to pee now" etc.
something that can help with that is pick a day where you have plenty of time, go to the toilet to empty out then put the nappy on and wear until in needs changing then actually change into another don't just take it off and go back to pants. do this often enough and your body and mind get used to it.
It's a similar thing with masturbation, I used to have the same issue in that I would get all horny, deal with that and then immediately want to remove the nappy. Also if I was thinking about putting one on and had a little "private" time first I would no longer want the nappy after.
There are 2 basic solutions to this which can be combined to a degree, the first is to try not to satisfy the horniness do other things or try to ignore it whenever you are wearing this will after a while disassociate the nappies from sexual feelings.
The second way though is to not be ashamed to feel horny with the nappy on, there are a lot of sensations, pressures and and other elements that will erotically excite you with them so if ignoring it doesn't work then let yourself have some fun and then try to keep yourself from removing the nappy, after you finish just lay back, breath and enjoy the feelings focus on how good it felt/feels eventually the "guilty" or "dirty" feeling will diminish.
All of it really just comes down to giving it enough time, persistence and wearing more.