All signs point towards this being a problem with the oscilloscope itself. It appears that when you change timebases from 25 μs/div to 10 μs/div, the sample rate also changes from 2.00 MS/s to 5.00 MS/s (the scope says so, on the top line of the screen). Clearly, there's something wrong with the vertical circuit that only shows up when the ADC is sampling at 5 MS/s. This could be as simple as a bad capacitor on power supply, preventing the ADC's supply voltage (from which the reference voltage is surely derived) from recovering fast enough at higher sample rates. Electrolytic capacitors do tend to go bad with age, so I wouldn't be surprised if this is the fault.
Fortunately, the TDS640A is from the era where you can expect a detailed service manual with troubleshooting flowcharts to be available, and indeed one is, right here. Try following the troubleshooting steps there (starting on page 225) and see if that can't lead you to the problem. If you have a computer with a serial port, or a USB-to-serial adapter cable, you may be able to get the scope to just tell you what's wrong with it (very fancy, for the time!). The troubleshooting steps all seem to end in "replace the X module/cable/board", which is likely no longer possible to get easily, but it shouldn't be too hard to hone in on the source of the problem with the X module/cable/board and replace the faulty part.
Edit in response to some comments:
At this point, I'd recommend you just open up the scope and replace all the electrolytic capacitors in the thing, outside the CRT drive circuit at least. Electrolytics from the 90s are not going to be in good shape some 20+ years later; they degrade with time, whether used or not. It should be fairly cheap and easy to get equivalent capacitors today from Digikey, Mouser, Newark, or whatever supplier you prefer. (I implore you not to get them from Amazon, Ebay, or Aliexpress, though.) I recommend going with capacitors from a recognised manufacturer like Nichicon, Nippon Chemi-con (aka United Chemi-con in the US and Europe Chemi-con in Europe), Rubycon, Panasonic, or Würth Elektronik; the very cheapest ones from no-name manufacturers tend to have quality problems in my experience.
I specify not to replace them in the CRT drive circuit not because those are less likely to need replacing or anything (they're not), but because working on a CRT is dangerous. The whole circuit is enclosed in a separate box with high voltage warning labels for a reason--lethal voltages can and do remain on capacitors (particularly the CRT itself, whose anode can store lethal amounts of charge at upwards of 10 to 15 kV), and should absolutely not be opened unless you know what you're doing--and should almost never be opened when you do know what you're doing, either.
