I love Copenhagen in August. Amsterdam is my second favorite city in Europe (after Paris). So it's hard for me to say.
This might be a tie-breaker -- Copenhagen has this thing called a Copenhagen card that allows you to travel around not just the city but the entire region plus gets you admission to most museums and tourist attractions. Including, if you are interested, the Louisiana museum a little north of the city (easy day trip by train) which is wonderful. Since you are only going for 5 days, this would greatly simplify your trip and also keep costs down. Also another easy day trip from Copenhagen is Malmo, Sweden, which is also a short train ride and has a slightly different vibe.
But Amsterdam is truly wonderful, and really great for city wandering, the sort of shopping you describe, and great cafe culture (might give the food in Copenhagen overall a slight edge if you like inventive/experimental cuisine -- check out the Michelin guides for both cities but Copenhagen is better known for its cuisine).
Sorry if this didn't help you decide. Either one is a great getaway for what you describe! Personally I'd try to fly direct. Check SAS, sometimes they have very affordably priced tickets, especially if you are traveling someone light.