Have you thought about getting a CD writer?
The most painless way to do this would probably be using a Squirrel PCMCIA-scsi adapter with a SCSI cd-writer, as this would mean you could unplug this and plug it into all your amigas easily.
By writing to CD you can also store them on your PC easily, or upload them to a web storage area (ie keep them safe against future hardware problems).
This would also mean should you decide to use WINUAE there would be no problems in transferring your files to your PC.
The hardware would also allow you to easily install OS3.9 (should you so desire).
Hope you get things transferred easily, oh and welcome to Amiga.org :-)