I also don't think you are a 1950s man.
It's pretty much the reverse in my house hold as I work PT. So it makes sense I do those things when I work 3 days p/week and DH often has 7 days on the farm.
Anyway how would a 1950s housewife have tackled these issues:
I organise vets, deworming, etc., of the dog. - use the landline to phone, write a letter or make an app in person
I do all/most of the organising of stuff (deep cleaning/sorting STUFF in the house). - this should be him organising. He can (like I do) ask for help with some big tasks that are easier with 2 ppl.
I do all driving (fair amount, with all kids' stuff) as he doesn't drive. - walk
I do most of the general arranging of stuff (for kids, for the house, etc), as he doesn't have a mobile phone. - again landline, letter or in person
I organise dentist/doctors' appointments for kids, and mostly take them. - above
I mostly take the kids to their clubs, etc. (see driving point). - walk
I do most of the financial stuff as he doesn't do online banking. - go to the branch (I know that is not very helpful anymore!)
I do most of the online shops, and all the larger shops in person (see previous point about driving). - walk/ take public transport to the large shop, he can arrange to be finished for when you will be passing after work to drive him + shopping home.
OK I am probably being a bit harsh on some of these. But really he needs to get a mobile phone. Walk the kids to their activities. Do online food shopping.