Is there any way you can take just a carry on bag? I went to Germany for 3 weeks a few years ago with one carry on bag. I haven't checked baggage in years and it's totally worth it. If the bag doesn't fit in the overhead bin or the bins are all full, they just stash it under the plane and then you can pick it up at the gate and remain in the airport.

If I remember correctly, the SLC airport is pretty small and the baggage claim is outside of the secure area.