You're finding why guys doing long exposure shots use a tripod! A longer shutter speed requires the camera to stationary during the exposure otherwise you'll get the blurs and streaks (as you've found).
Aside from using a gyro stabilization rig, you might try exposure bracketing which your camera can do. I think you can get 3 bracketed images per shutter release at up to +/- 3 stops of exposure on EOS models. This is an HDR technique, but please don't. HDR is WAY overdone. Instead in photoshop you can mix elements of different exposures to get the pic you want.
When you bracket, make sure the shutter has priority so that each exposure is the same length, then the camera will overshoot and undershoot the exposure with the aperture. RTFM.
It should go without saying, but I'll risk saying it. Keep your on-camera flash off.