This really would seem to be a problem with the ASPI driver files you have (winaspi.dll, apix.vxd, wnaspi32.dll, aspienum.vxd). Most likely the conflict/problem is with apix.vxd and/or wnaspi32.dll. Your EasyCD installed an Adaptec set of these files, but your version of EasyCD is really old and as a result the aspi files are really old too. The most current version of those files is 4.70 (but 4.60 would be fine).
In any case you'll likely need to do one of two things: 1) try and update the aspi files and see if that solves the problem, or 2) replace the files APIX.VXD and WNASPI32.DLL with the original MS supplied versions of these files off your Win98 CD. I would try to update first and if that fails then revert to the MS files.
To update your files:
1) Download
this file and unzip it to a temporary directory.
2) Boot your PC to a dos prompt (don't go in to windows and don't use a dox box within it), and switch to the directory where you unziped the file above.
3) Type in dumpasi. This will create backups of your existing files.
4) Next type in instaspi. This will update the aspi files. When the bat file is done you'll get a message about rebooting your PC. Reboot and let windows come up.
After doing the above see if your problems are resolved.
If you still have problems then you will have to use window's SFC to restore APIX.VXD and WNASPI32.DLL from your win98 CD. WNASPI32 needs to be restored to the Windows\System subdirectory and APIX needs to be restored to the Windows\System\Iosubsys subdirectory. Once these files are restored then reboot and see if things work this way.