Let's keep this thread constructive for those that will have a similar problem, even if the original poster has jumped ship to buy a new house.
Sewage gas smell in a home is often hard for many homeowners and novices to diagnose, mostly because none of us have x-ray vision to see pipes in walls and floors.
I'm going to write this article in order, starting from least expensive and easiest to do.
Stopper block all drains. One visit to a hardware store, and come back home with a collection of rubber stoppers, and plug them all. Buy as many as you can, keep your receipt and you can refund the ones you don't use. Yes, you'll have to remove a stopper manually each time you wash your hands, but at least you can breathe without short breaths and dizzy spells.
Health is the most important here, and this will give an instant improvement to your quality of life. Finally, with a clear non-dizzy head, you, and certainly other paid professionals that won't quit on the job, can finally investigate.
Make sure all the p-traps contain water. I like my house to have p-traps that can be checked anytime without tools. I would replace all p-traps with ones that have a very large plugs that can be removed and put back on by hand, without any teflon tape. ABS is soft to cut, and cheap to glue. If a good amount of water drops out of your traps, confirm all of them contained enough water to stop sewage gases. A good thing to use is soda pop you are familiar with, so you can still smell it in the trap, or get a contrasting different smell with sewage.
If any traps empty out by suction, there is a blocked attic vent, this usually happens at the T or Y where the sink plumbing meets up to the rest of your house. Cut that off, and you will see it is full of crud. You might not be able to clean all the decades of silt dirt leaves snow ice dead birds squirrels that crawl in. Close up the repair with new ABS and install an AAV (Air Admittance Valve) for that sink. Despite all the AAV bashing, AAV's are something that will solve your problem temporarily and give you time for a permanent solution. I can install an AAV in 15 minutes in Janurary, get my health back, and then worry about tearing out the attic vent pipe in Spring. Most folks that tell you not to use AAV's usually aren't dying in sewage gases while reading this post, you are.
Now you have solved and possible venting problem and see your traps are always full, next is to check your toilets.
Remove all your toilets. Holes from missing wax is common from bad installations. That can lead to loose flanges, with entire floors missing from decades of rot. If the flange is confirmed healthy and doesn't need repairs, you're going to get new wax and reinstall your toilet.
Still got sewage? Then you have a break somewhere in the wall or ceilings. Check your attic. Vent pipes that run vertically from roofs can fracture and split. You will have to start cutting rectangles in your drywall to see your pipes. A purchased camera and leaving 3/4" holes might be less intrusive, but the only problem with cameras is that you can easily miss entire areas. Whole strips of drywall removed to see the entire pipe lets me see the whole pipe to be sure. Your health is on the line here, and repairing drywall is cheaper than buying a new house.
If everything above the sewerline checked out, now it's your concrete floor in the basement. Confirm without a doubt that process of elimination tells you do not have a problem anywhere else. A professional will camera your sewerline from your floor to city services. These guys usually aren't your regular handyman plumbers, they are used to troubleshooting commercial installs over long pipe runs. Not all old homes have clean out traps, and there will be some digging to install one. With a new basement clean out installed, at least we can close it up while you decide on how to get a contractor to rebuild your sewage to the city. I guarantee they will find a problem here, assuming you've confirmed isn't anywhere else above your basement floor.
Drain pipes in a typical residential home aren't mytified, eventually we can get a the whole picture with zero guesswork. The problem has nowhere to hide.
If you think any of the above is drastic and extreme, feel free to stop and just buy another home.