Start looking for a second hand pure analog CRO in good condition. 50 or 100 MHz bandwidth, dual channel, delay timebase. If you can find one with a linear PSU, and no digital readouts, it will emit very little EMI. Don't get one with a storage tube as they've nearly all aged to the point the storage mode is unusable. Ask here for advice on specific models before buying to avoid known lemons.
You'll still want to keep and use your DSO because its far more versatile and convenient, but if you need low EMI an old-skool CRO is the way to go if you don't have a five figure budget. However once you start chasing the elusive low EMI environment, you'll find just about everything in modern life is problematic, from the LED or CFL lightbulbs overhead to the SMPSU wallwart down on the skirting board, including the phone in your pocket. If you live in a dense urban area, its pretty much hopeless due to all your neighbours gadgets.