Don't forget the 1.6x multiplication factor! At 300 mm you have a 480mm lens, effectively. To get sharp pictures while holding the camera in your hand, a shutterspeed of 1/500 would be nice. I have the same 75-300 lens, and was browsing some pictures. Indeed the ones with the shutter speed around the focal length were the nice sharp ones.
Cranking up the ISO setting will get you the higher shutter speed. Given enough light (and aperture) you should be able to get nice pictures at ISO 100.
Maybe I'm totally off, but who knows