Rats need to live in a large cage.
They need to be in a well used room in the house so they get a lot of social interaction - and should not be kept in a shed unless it is fixed up and you plan to live out there with them.
They can not live in a closet and certainly not in a freezer.
Air circulation is essential to their health, and as mentioned above, ammonia will build up and destroy their lungs.
Rats also need to be around you and interact with you for several reasons including so you will notice any small changes that could indicate that they are ill.
When rats are in a room with a great deal of air circulation, such as an open living room then the smell etc is much less then when they are shut in a room. I found a huge difference between living in an apartment where they lived in the living room and renting a room where there are confined to an enclosed room.
An air purifier with a permanent hepa filter and a replaceable carbon filter (not an ionizer) will help, but is an enclosed room I don't know if it will be enough. Make sure you buy one large enough to handle the room it is in.
If they are to stay in your bedroom, not only will you need an air purifier but an air exchanger for air circulation would help if you do not already have one.
It would likely help if the cage was cleaned daily with bedding and hammocks being replaced - but not too thoroughly as if too clear then they will urine mark even more.
How old are the boys? From what you wrote I assume they are young babies.
If so, I am surprised that you are reacting this strongly to them.
Is it possible that you are allergic to the bedding that is being used?
Many people are allergic to the bedding and not to their pets.
Unneutered boys have 100 time the amount of allergen in their urine then boys who are neutered so if you have access to a good vet with the knowledge and experience to safely neuter rats then you might consider doing that. Only gas anesthetic is given and nothing is injected until they are given metacam after the surgery is over. Most vets do not know how to neuter rats safely. If a vet does not have the necessary knowledge and experience then there is a good chance the rats will die during or after surgery.
https://www.ratshackforum.com/threads/why-spay-or-neuter-rats.35402/
I have been getting my rats spayed and neutered for the last 15.5 years.
Long sleeves, washing hands, keeping the cage clean, etc are all things that can help. As well as having the boys neutered. Jorats can likely tell you how she dealt with rat allergies for many years.
From what you wrote, I don't think you should be sleeping in the same room as your rattie boys.