For some, playing sixteenth notes at a brisk tempo while double tonguing can present coordination issues as you described. However, at this point in developing your playing skills, I would suggest NOT using trill fingerings or alternate fingerings at this time. There are times when trill fingerings are appropriate, but you are looking for a short cut when you should be concentrating on developing and improving your technique. Instead, use a slow tempo and practice those passages using the normal fingering while double tonguing. Use a metronome to keep a steady tempo and practice playing those difficult spots as slow as needed to play it cleanly while double tonguing. When you can play it cleanly 5 times in a row, then move the metronome tempo up one notch faster and practice it until you can play it cleanly 5 times. Repeat this process until you can play the passage at or even faster than the indicated tempo. After a while, you'll find that you can play those measures without a struggle. It might take a few days, a few weeks, or even months. It really depends upon the individual. Some are able to progress faster than others. The main thing is to take the time to learn to do it correctly. Then, when you encounter even more difficult music, you will be better prepared to know when to use the trill/alternate fingerings for something other than trills.
"Never give a flute player a screwdriver."