I'm not sure what brand of flooring you have or what cleaner the store you bought your flooring sold you. The brand I've used is Pergo, I picked mine up at my local home improvement warehouse. It has alcohol in it, which means it shouldn't leave streaks like your current brand of cleaner and it's all natural so you know it won't damage your flooring. As for the paw and foot prints it's hard to tell if that's just the nature of the flooring itself because of the type of laminate/coating or if it's something that you should be able to get rid of using the correct cleaner. Have you tried contacting the company that makes your floor? If I were you, I'd try to get in touch with them and see what their technical support suggests to use.
The Flooring Diva
www.theflooringdiva.com