Here are some ideas;
1: Record your amp by putting a microphone in front of it. You would also need an audio interface or at least an adaptor to connect the mic to your computer. Before I started using an dedicated interface I had a small Behringer 5 channel mixer connected to my sound card input. That worked.
2: If your amp has some type of line out jack you could put a DI box with an attenuater switch in between. Live Wire Solutions SPDI has worked for me. Or use a Behringer GDI21 which has great reviews.
3: There are many decent digital FX pedals that do a decent job of amp, over drive and distortion emulation. My current choice has been Zoom G1 Four on the Fender Deluxe Reverb setting. The output on the G1 Four is a single mono or stereo output. I can record straight to my sound card (although I often don't) with a simple cable and an adaptor at one end.
4: Record direct as you have been and use any one of the many and often free plugins such as:
SimulAnalog (various plugins)
Voxengo Boogex
Silver Spikes Ruby Tube
Acme Bar Gigs Whisper Tube
Blue Cat Audio BC Free Amp (2 thumbs up for this one)
MSaturator by Melda Productions (can adjust 2nd and 4th harmonics)
Amplitube 3 or 4.

Does the noise in your head bother me ?