Hand galloping is a useful addition to your riding repertoire and a good measure of your riding ability where you allow the horse to let go a little but you stay in relaxed but effective control.
When training your horse to change to a different discipline requires specific attention to riding details. if you are softening your horse down then dressage is the obvious choice but professional help may be necessary. Here are some tips.
A correct horse training method must include being sensitive to the feelings of your horse. Keep the training varied and don't bash on with the same exercise when it is obvious that he is no longer interested. This excellent video discusses the best way to train your horse for maximum results.