continue running animation sequence and run main loop again to check for strings (if statements)
There is still the assumption, incorrect, that the client can send more strings. It can't. It sends all the data needed to satisfy one request at one time. The "animation" you are looking for comes from the same client making multiple requests in the proper order with minimum delay between requests. There is nothing that the Arduino can do to get multiple requests from the same client any faster than the client makes requests.