@YourVividDuck - firstly, condolences re your dad. Am so sorry you’ve found your way to this board, it’s a place you don’t want to have to find, but everyone here is lovely and understands how you’re feeling.
It’s definitely not stupid to be feeling the way you are re going through his things. It’s heart-wrenching as am sure you want to make sure that you don’t miss a thing but, as you say, it feels an invasion of their privacy. I really struggled with those feelings, but just tried to tell myself that mum loved so much and that, if anyone had to have full view of her life, then she’d trust me to do it and to do it thoroughly and respectfully. It’s a task that is incredibly hard emotionally, but I try to remind myself that the reason it’s so hard is because of the love that I feel for mum.
Everyone handles things differently, and sometimes you can feel a sense of pressure that you need to get everything “sorted”. As long as the important/legal things are in process, don’t feel you have to finalise everything. It’s perfectly ok to, say, keep some stuff aside to revisit a bit further down the line.
For now, just try to be kind to yourself. Grab all the sleep that you can, try to eat something each day, no matter how basic or lazy your choices may seem. And take any offer of support that’s available. And post here - there are no stupid feelings or reactions for what you’re going through, just a lot of people who “get” it.