If I awake in the middle of the night and can’t fall back asleep because I am particularly worried about something, am I allowed to recite Tehillim? Do I need to first get out of bed and wash negel vasser or say burkas hatorah? Or can I just say it lying down as I try to fall asleep?
You should preferably wash your hands before saying any davar shbikidusha, but if it hard for you, you can just wash a little of even wipe your hands with some cloth etc. because you most probably touched a covered part of your body while you were sleeping.
You don’t have to say birkas hatorah. This is because you are not saying the tehillim for learning sake, rather you are davening, and for that you don’t have to say birkas hatora.
H-shem should help you and all the things that are worrying you should work out well
Shulchan Aruch O:CH 46-9, 4-23 M:B 60,61, Ohr L’tzion Chap. 2-1