Oh OP, I am so sorry.
My husband's mum (my MIL) died on Boxing Day. It wasn't in the same circumstances as your dad, she was ill, but it is still hard to deal with.
We have been going through our own grieving processes for a few days, and it is already feeling different compared with the first day or two, things will change over time. At the moment, what I can say is, the grieving process is different for everyone, so what you go through may be different to what siblings, aunts etc. go through.
Don't compare your thoughts and feelings with others and start thinking you are not 'doing it right' and feeling worse about it. There is no one way to grieve, with us, DH felt numb for a couple of days, I cried a lot.
Don't be ashamed to cry or get angry etc. they are often part of the process.
Don't be afraid to ask for help or ask people to help organise things e.g. parts of the funeral service when the time comes.
DH is organising the cremation, he was a minister for a few years recently, and his mum wanted him to do it. The funeral director is available to help and guide him, (not just organise the service etc.) they can't be there at the snap of our fingers, they have other people to help too, but will help asap. If you have no idea which funeral director to use it might be worth your while to ask a few people who have been through burying a loved one who they would recommend. In my experience established funeral directors are usually really good at their job, they tend to have a lot of experience, empathy and compassion etc. (It tends to be natural with them, it is hard to do a job like that without being naturally inclined towards dealing with the bereaved). A funeral director can help with guidance as to what will happen even before post mortems are done and bodies are released. They have a lot of knowledge with how things happen in a huge array of situations. If you choose a funeral director and they 'don't work' for you, you are allowed to use a different one.
You must also look after yourself. For the first day or two DH hardly ate anything so I had to make sure he was OK. It would be no good for him or us (4 DC) if he made himself ill. If things get 'too much' for you, you need help. If you need to talk, find someone to talk to, don't keep it all inside. If you need help from a gp, make sure you get help. Hopefully your family (and friends) will be there for you, not just you being there for them. There are plenty of us here on MN for you too. PM me if you feel like you want to or need to. I won't know all the answers, but will try to help. Even if we just have a cry or a rant together.
Sorry this is so long, there is so much to go through and sort out though. I hope you eventually got this far and hope you get everything sorted.