My husband and I did the Southern Tier east-to-west this spring, and ending up skipping both the Davis Mtns. climb (planned) and Emory Pass (forced by terrible wind conditions to alter our route). We did, however, see both climbs by car and thought that the Davis Mtns. climb looked much worse than Emory Pass (which seemed more long than brutal). To avoid the Davis Mtns., we stayed on 90 through Alpine and on to Marfa. We did the ride from Marfa to Van Horn on 90 in one afternoon -- nearly 80 miles (long for us). It is nice and flat and not a difficult ride. The only services are a dilapidated grocery in Valentine (but don't count on it still being open).
Our shortcut around Emory Pass took us on 1-10 from Las Cruces to Lordsburg over two days. There are few services on 1-10 on these days -- one "trading post" on the first day until you get to Deming, and then a couple of "trading posts" that have Dairy Queens between Deming and Lordsburg. Then we took US-70 from Lordsburg to Safford, AZ, rejoining the official route just outside Safford. The bonus is that US-70 is an amazing ride -- almost no cars, great shoulder, beautiful scenery, and a town (Duncan) with a couple of restaurants and a small grocery about halfway.
Check out our journal at http://thebigtrip.crazyguyonabike.com
if you want more info, or email us. Good luck!