In addition to the body leads and signals coming through the arms and hands, signaling happens with subtle movements of eyes, head, etc. As women gain dance experience and knowledge of their partners, they tend to pick up the finer signals and respond without even thinking about it. A good follower will compliment the dance style of the leader she is dancing with. Keeping your mind ready and open so you can follow takes practice. It is particularly difficult if you have had a disturbing event happen immediately preceding your dancing.
Balance is very difficult to teach to a student who lacks it.
Dancing independently until you are capable of keeping your body moving easily, and wearing sufficiently flexible, well fitting, comfortable dance shoes will help. The best way to keep from sliding when coming onto a foot is to already have the body weight over the foot when the foot stops moving. In other words, when you step forward, move forward with the body, rather than sticking the leg out first. Then, just place the foot underneath where the body ends up. In her workshops, Beth Emerson recommends committing your weight immediately each time you step on a foot rather than having your weight split between feet for a moment, which is what we tend to do when walking.