Macro needs more light than usual. This is the source of all your problems. The IS in the L lens may help, sometimes, but it's probably better to work on fixing the lighting.

At close macro focusing distances, you need to use a small aperture (bigger "f" number) in order for the whole object to be in focus at once. Unfortunately, this reduces the light reaching the sensor, requiring a longer shutter speed.

At close macro focusing distances, the slightest camera movement causes significant subject movement.

Handholding the camera, still enough to not be noticed in a macro shot, for the longer shutter speed required is pretty hard, as you've found out.

There are two fixes. One is to eliminate movement, which the tripod solution. The other is to reduce the shutter speed by adding more light. This can be bright constant light sources, or a Speedlight. You'll want to get the Speedlight off the camera, because at macro distances, the light from a Speedlight in the hot shoe won't reach the subject. The light doesn't shoot down, and the lens is in the way. You can fix this by reflecting light (mirrors and white poster board), or by adding a trigger on the camera, and a lightsource off the camera. Canon's cheapest 90EX flash can actually act as a master, while all the other mid-range flashes, up until the 580EX/600EX-RT, can only act as slaves.

Long story short, use a 430EX off-camera to light your subject, and use a 90EX to go on the camera to trigger the 430EX. Put the camera is M mode. Select a shutter speed of 1/200s, ISO 100, aperture f/whatever you need. The camera will magically figure out the amount of flash required to correctly expose the shot.