I used to have this problem, but then I performed a little fix outlined in a different thread. User Haku figured out how to set the default path for your save states to the sdcard rather than internal memory.
For anyone in the future who has this issue, here is what I did specifically:
-On the SD card, created a separate folder called "PS1 Saves"
-Opened up FPSE and started a game
-Once the game got going, I selected system settings, then Savestates Path
-Found the "PS1 Saves" folder on my Sd card and selected it
-Backed all the way out back to the game
-Open the UI Menu in order to select Quit
-In the submenu for Quit, select Restart
-Saving and Loading should work
*Might take a few times to work as of this version
After I did this I could save and load without crashing. However later on as I was messing with some settings I experienced it again. I think it was because I was tampering with HLE mode, so my advice is to not touch HLE.
Once I performed the fix listed above an found my desired settings and saved as default, saving and loading works as it should. Just keep in mind that save states remember the emulator's settings at that time. This might attribute why it's crashing as well.