Normally, if you saved by talking to the floppy disk at the Looong Gallery, the title screen has a 'continue' option.

I'm not sure if you're playing in the browser or with a downloaded copy. If the browser didn't save your game, you might have some browser setting that is clearing your cookies. 

I really recommend downloading the game and playing on your own emulator. I'm sorry to hear if you lost some saved progress.