Wow - what a great analysis! Kudos
Sounds like a program or service with bad memory management. Does Task Manager show any programs running while the Handles count increases? If so, which programs?
See if you still have a swap file set up after "running out of hard drive space"
- Start>Control Panel>System>Advanced tab>Performance-settings>Advanced tab>Virtual Memory-change>Choose "Let Windows manage" or "System manage" ->Set, then "Okay" out. Re-boot.
See what happens to the Handles count if you boot into Safe Mode. Likewise, in 'Normal" mode, try this: Start>Run> type
msconfig, enter>on the General tab, select the 'Diagnostic Startup' option, "okay" and re-boot.
Advise back.
EDIT: Dias and I posted at the same time - start with an av scan, if you haven't already.