Having dogs run on pavement might not be good physiologically, and having dogs run in traffic is crazy. I wouldn't do it.
There are probably some trailer designs in the Commuting (
https://www.bikeforums.net/commuting/) and Utility Cycling (
https://www.bikeforums.net/utility-cycling/) forums. I know there was one preson (maybe @
CliffordK?) who hauled a lathe home---some quarter-ton of machinery---but he had a specially built bike.
If you want to go car-free and haul 180 pounds of dog safely , you are going to have to spend money. Face it. Save for it.
This is serious. If you rig something and your rig breaks by the side of a busy road , you will have two 90-pound dogs---much more energetic than t=children and probably a lot less obedient and intelligent than a couple 90-pound children--and the dogs will be much more instinctual. They won't understand what is up ... but they will have strong urges to run, bark, jump, investigate .... all of which could endanger them or others.
Not trying to be negative, but breaking down on the roadside with a month's worth of groceries is bad enough. Breaking down with a couple of large, powerful dogs is very serious.
I'd say ... just a a parent with children ... you might be in a situation where going completely car-free is just not a reasonable option. You have responsibilities to other living beings which Must come first.
I would suggest investigating a trailer/towing rig which can safely and reliably get your dogs to the park. They need it, and for you to deprive them of quality of life even to improve your own quality of life is unfair. Wait to go car-free until you can provide what each of the significant others in your life needs--the SO And the animals.
If you have to save up for a year to do right by th4e dogs ... I Hope they are worth it to you.
I am not sure you need to spend $1200 on a store-bought trailer. I know there are things you could build or modify. 200 pounds (allowing for snacks, water, towels, and a rain cover) isn't a Huge load. It is about three times what most r BOB or Burley trailers are rated at .... but I have a friend--not even a cyclist---who crossed the country on basically Wal-Mart bikes on cobbled-together trailers towing about three times that. He supported himself mostly by busking---playing music at bars occasionally---so he wasn't wealthy. His rig wasn't super-reliable, but it was pretty good--and you would be hauling a lot less.
There are ways you could haul the dogs safely. There are probably ways which cost less than $1200. Start researching, (the two forums are full of people eager to help) and start saving and may all the right things work out exactly as they are supposed to.