Im so sorry for your loss. I know just how you feel. My dad had something similar to motor neurone disease and was very dependent. He died from pneumonia in June.
I would say that the hardest bit is definitely after the funeral and it left a massive gap in your life, not just emotionally but physically because of the necessary routines, visiting, errands, sorting medical appointments and generally being an advocate for Dads needs. That takes a long time to adjust to and it does feel strange and, yes, even wrong to suddenly have a daily routine that seems so 'free'.
Like you say, it's very very early days and you need to be kind to yourself. It probably goes against the grain to do this, but you need to allow yourself to grieve regardless of the needs of others. You are not responsible for everyone else's comfort and believe it or not they will survive if you're out if action for a while. You might even find that your dh steps up if he sees that you need time and space to just let rip. Try to stop focusing on other people's needs and take care of your own, otherwise you might find it is much worse at a later date.
I found counselling really helped. I don't know if your dad was helped by a hospice but mine was, and they have an excellent support service for families way after your loved one has gone. If not, your GP can sort bereavement counselling; or try CRUSE or maybe there is an organisation that supports your dad's condition? The counsellor helped me deal with all my feelings of guilt, just like the ones you mention.
An excellent analogy she gave me was to imagine your emotional wellbeing can be represented by a stockpot. Usually, you keep topping up a stockpot every time you take from it. For all the time your dads been ill, plus you've kept the family going, your stockpot has been running down but it's not been topped up. Now your dad has gone, you need to focus on filling up your pot again - and you can only do that by letting yourself grieve and looking after yourself more than you look after others (apart from your dcs, although your dh could help with that as well, I hope).
It took me a fair while to get used to saying no to anything or anyone's needs that were not absolutely vital but I'm sure it helped. We went on a holiday away from everyone and I literally stared into space for 2 weeks shortly after dad died. I know this might not be possible but you do need to recuperate and not commit to anything or anyone.
I hope you can do this. All the very best. Dont rush it either; it will take time. X