Sound shouldn't leak that much with open back headphones, you're probably listening too loudly, especially when outside as you're trying to compensate with the external noises with your music.

Try not to raise your volume before you leave your home as yes it's damaging your ears when you make it compete with outside noises.

I have the Sennheiser HD 560S too and I can power that (one-sided only, lol) with the FiiO UTWS3/5 TWS adapters just fine and it's yes easy to get high volume, so yup don't try to increase it so much when outside.

But yes the HD 560S can leak quite a bit with certain albums/songs/etc. even if the other ones in your playlist don't leak at all. It's just varied like that and so you have to know which ones are more likely to leak, like usually you'll know as the volume is louder (even if you didn't change the volume on the app/phone/anywhere) but sometimes those songs are after you're already listening for a long time and our ears get easily used to loud noises and so it's easy to forget.

So yes just be mindful if you don't want to bother the other people commuting on the train/bus. Though I must emphasize again to try not to raise the volume once you leave your home as even though you likely won't be able to hear anything against the bus engines, people's conversations, et cetera, it's the safe way for your years.

Try IEMs or TWS earbuds instead for the better passive noise reduction. Closed back headphones, especially those with ANC/Bluetooth will also work well with commute.

There's Bluetooth amp/DAC dongles like the Qudelix 5K btw, but make sure to search up on how to use them with Apple/Lightning devices as people need special OTG (allows data transfer) cables and so on. Or adapters/converters like the Apple camera kit or something like that, so that the dongles work properly with the Lightning device.

Those Bluetooth amp/DAC dongles usually have parametric/etc. EQ these days and so you can easily change the sound there. As yup, covering the grills of open back headphones are not gonna really change the sound to what you think. Like usually it doesn't do much even with the noise reduction/isolation.

Try using aftermarket earpads like Dekoni, Brainwavz, etc. instead. Look for the leather/etc. earpads, or other materials known for noise reduction as velour is usually there for the comfort/sound.

Actually parametric EQ will have easily noticeable changes too.

Don't forget to check the subreddit, /r/oratory1990, for preset parametric EQ list (mainly for headphones but there's some IEMs there): https://www.reddit.com/r/oratory1990/wiki/index/list_of_presets

See here for more info about parametric EQ, squig.link comparison graphs, AutoEQ, et cetera: https://www.reddit.com/r/headphones/comments/109meb5/eq/j40cy1n/

Open back headphones noise reduction (scroll to the bottom half part of the comment), earwax clogging issue, parametric EQ, noise reduction for your PC/room, et cetera: https://www.reddit.com/r/HeadphoneAdvice/comments/115c99b/coming_from_iems_and_looking_for_headphones_150_eu/j92mpe9/