Hi Rachel. I took my Canon DSLR + lenses to Europe for quite a few trips before I got tired of lugging it all around. (not mirrorless, but my DSLR has always been prone to dust on the sensor also - drove me nuts!!!!). In 2017, I finally gave up on it for travel and instead bought a "bridge camera," a compromise between a DSLR and a P&S. I got the Panasonic Lumix DMC-FZ1000. It has a fixed zoom lens, the equivalent of 25mm-400mm. It has limitations vs. a DSLR for sure, but at least I know what they are, and mostly I have learned to work around them. I've been surprised that I really do use the lens both wide and tight and in between. (If I could choose, I'd love to get a little wider.)
Anyway, I'm not sure I could live with just one of my DSLR lenses. This camera has been a great trade-off for me. (Still need a tripod though - wouldn't give that up, because I love taking night shots.)
Sony has a similar (some would say superior) camera to mine, which is a bit old by now anyway - DSC-RX10.
As far as memory cards: I have one 64GB SDXC card in my camera, which is usually enough for a day's shooting while I travel - I shoot every frame as RAW+JPEG. If I shoot videos too, that will really eat the space quickly, but I mostly shoot stills.
I always travel with my small laptop plus a portable hard drive for backup. Every night, I dump the memory card to my laptop and then backup all of those photos to the backup drive. That way, have at least two copies of everything, on separate media. And once I'm done that for the evening, I format the memory card in the camera so it's fresh for the next day's shooting.
I often wind up going through my photos while I travel too and can do soon the laptop.
If I didn't have the laptop? Then I'd buy a bunch of extra memory cards many more than really needed so I could swap them out every few days - they are pretty cheap nowadays. And I'd try to figure out a way to get some of the images (JPEG, anyway) over to my phone, to have at least some of the highlights saved in case of a catastrophe, so I'd have SOMETHING if I lost my memory cards or something.
Having at least one spare battery for the camera is essential also. I tend to buy cheap after-market batteries on eBay or Amazon for my spare batteries. They don't have the same capacity as the original, but they cost a fraction of the original. If they only last 2/3 as long, doesn't matter because they probably cost 1/5 what the original cost.