You can’t do grief wrongly.
What are you comparing? What’s ‘falling apart’ vs not falling apart? Normal life has to continue, we all have to deal with death and then go to work, deal with loss and then cook dinner, grieve and read a bedtime story, cry in the shower, lean on friends at social events etc. And you will too.
Most of the immediate aftermath of death is all practical and there’s no way round the arrangements, the phone calls, the will, the funeral, the celebrant, the invitations, the notice in the paper, dealing with their place of work, booking the food and the service and the readings and the outfits, the other family members and their feuds, the utilities, the house, the life insurance, their stuff, the pensions, it’s endless. So everyone dealing with death suddenly has to deal with all of those things in addition to normal life and going to work and looking after DCs, I wouldn’t call that falling apart. I’d call that having a shedload more on your plate than you did a few weeks ago.
im sorry for your situation and hope your DM finds peace x