Again, I defer to the veterinarian that has examined your pet, he can give you the best estimate of age and nutritional requirements for your dog.
Did you read the article (link) I sent you?
That’s the best advice you will get from a veterinarian that has NOT examined your dog.
I for one don’t bother with “puppy food” and go straight to adult.
The large breeds puppies needing this and that is a new thing. Discuss with your vet.
Good luck
PS: “The shelter told us that we could switch him to adult food after he finished the mix of adult and puppy that they sent him home with”.
Sounds like sage advice.
Fromm Classic Adult is a good food (my vet agrees) https://www.gofromm.com/fromm-family-classic-adult-dog-food
Add a topper ( a tablespoon of canned food or cooked chicken or scrambled egg) and a splash of water and you’re good to go!
Always have fresh water available 24/7
Oh, and please do not free feed (leave food down all day) that would explain the rapid weight gain, not good. Two meals per day, measured amounts, and an occasional snack is enough.
Exercise, start walking every day! Start brushing the teeth once a day, check YouTube for how to videos.
