Another comment mentions a KryoFlux. A cheaper alternative is a GreaseWeasel (£30, $30). The 4.1 version even has USB-C as a connector. You will need an external power brick with a 4pin Molex connector for 5.25 drives. You can buy these standalone or they might come with a SATA/PATA/IDE Hard Drive to USB converter. Can also use an old PC power supply if you fake the power on signal.

I've had quite a lot of success, almost 100%, over quite a long time - both 5.25 and 3.5. I did use older drives intended for "desktop" machines. The drives from Japanese companies seem to be the most reliable; NEC, TEAC. The 3.5 USB Portable diskette drives are based around laptop floppy designs. You won't be able to find a direct 5.25 to USB adapter (or modify one to work) as the speeds / data rates are all wrong, so you will need something like a GreaseWeasel or old PC for those.

I've used different reading mechanisms over the years; an old PC running Linux with a real floppy chipset + custom raw reading code, custom hardware with an F82077 + Atmel + SDCard, and most recently a GreaseWeasel.

The floppies I've recovered were stored relatively well, but were mostly my and family members data - not commercial floppies. If there is any copy protection on the floppies you are trying to read you may need to capture the flux transitions. For these, I'd check archive.org to see if someone has already generated a copy.

For 5.25 floppies fingerprints and surface blemishes can be softened / removed with ISO Propyl and a Q-tip. Floppies designed for cleaning the heads of drives seem to now be very expensive / non-existent. Manually cleaning the heads of a 3.5 drive is way more difficult. I've got ideas about using Face mask / filter material to create a cleaning tool, but it is on the experiments to try list.

Even back when floppies were a thing, I found that 3.5 drives were more un-reliable between machines than 5.25 drives. I think there was more variation in head alignment and track positioning. When I've had issues, I've always been lucky in that switching the drive vendor / model made reading the floppy possible even if it took multiple attempts or was very slow.

Good luck.