Short of finding an adapter or peripheral that allows you to transfer the saves from the memory card(s) to the computer itself (I have one somewhere for a PSx, don't work with PS2), you almost either need to modify the PS2 in some way to allow it to read / write save data to a USB stick.... Or find some program that does (Like the Action Replay max or just some memory manager).

Its kinda how Sony made it all priteritory (Of which they still do with Vitas, requiring special memory cards which cost twice as much as a micro SD card), they didn't want people to easily transfer saves around between devices...

That sucks, but will the emulators read those files?

I'm not sure if the emulator will or not. I know that ePSXe (Ps1) reads the PSx style memory card files that I have. The same might work for PS2 game files if the emulator is similar in coding as the real PS2.