I've recently bought a house that is about 20 years old. Within a week of moving in, I've found if too much water is going through the system (ie after I've showered, done the dishes, and a load of laundry), all drains (including floor drains) start overflowing. I've tried plunging, but it doesn't seem as if that's the problem. The house is also built on a slope near the coast if that helps, and was vacant 6 months before I bought it!
Abbey

You can stop plunging and start rodding the main near where ir exits the basement. Hopefully you will find a clean out in the basement since the house is only 20 years old. I always do a quick test but one of my plumber friends doesn't do it, now he does afetr flooding a neighbors basement with sewage. Before removing the clean out cap I get a 1/8" drill bit and a 1/8" self tapping sheet metal screw. Flush all the toilets and drill a hole in the cap, if water shoots out install the screw and you know the clog is down stream from where you are. No water, the clog is further back inside the home. Most main clogs are outside so definitely have that screw ready

Are you on a septic tank or city sewer? Let me know, Tom

Are you on a septic tank or city sewer? Let me know, Tom

I'm quite sure it's city sewer. I've had a plumber out, and he's said that it's necessary to dig up all down the side of the house to clear the drains, but I found him quite vague. He didn't do any testing (ie cameras down pipes)...

Get another two plumbers to give estimates and opinions. I would not hire any plumber who said what your guy said with a SeeSnake.

Run a Sewer Cam through the system and find out exactly what the problem is. What material are your drainage pipes made of? How old?
Most clogs on a city sewer occur at the street raiser. You might only have to expose the raiser to fix your problem. Get several estimates and go with the plumber you're most comfortable with. Good luck, Tom