I'm using a peak atlas ESR+ to measure the two large capacitors in a Commodore 1541 disk drive:
7380uf 0.10ohm esr 6800u 25v
4630uf 0.08ohm esr 4700u 16v
According to the chart in the peak book, for this size/voltage cap, the ESR values are double (4700) or triple (6800) what is in the table.
The chart in the peak manual is based on what typical MODERN series of capacitors are rated for.
You have capacitors from the 80s or 90s or whenever that unit was manufactured. Back then they didn't have quite as good formulations for electrolytes which results in higher ESR compared to equivalent capacitors from modern series.
You should desolder at least one lead of the capacitor to make sure that capacitor is not in parallel with other capacitors or resistors, which would affect the measurement - yes, you can measure in circuit, but the fineprint is "provided you know how to measure and be aware results won't be correct when other components are in parallel with the capacitor.
If you do decide to recap, consider using older series of capacitors which now would not be considered to have very low ESR. I'm thinking series like Panasonic FC or Nichicon PW - as a food for thought, Nichicon PW 6800uF 25v has 0.014 ohm esr, and same for Panasonic FC 5600uF 25v, highest 25v rated in the series.
Also, is there even a switching power supply powering the device, or is there a classic linear transformer power supply?
If it's a classic linear transformer providing power, then the ESR of a capacitor is really not a big concern, it doesn't have to be very low.