I agree both are fine, but have a subtle difference of meaning in BrE.
"I will be travelling from July to September" implies to me the idea of continuous movement throughout the trip, such as a student on a gap year travelling around the world. They may stop off at various points but the focus is on their journey.
"I will be away from July to September" could mean they will be doing some travelling, or staying in a hotel, or a mixture of both. You don't know exactly what they will be doing other than they won't be at home, or work, as implied by the context in which the message is to be set.