This is a great question. This is something I worked on when I was studying with Scott Lee, and I continue to work on now.
I think it's just as important to alternate between first and second fingers of the right hand on DB, just as you do on EB. If it feels awkward, just keep at it - it will feel natural before you know it. As you know, you are so much more nimble if you have more than one finger happening. It sounds like you just need to do it more.
Also, be sure that you aren't using ALL of your electric technique on the upright. A common struggle, for those of us who play both, is perfecting the difference between the angle of the right hand for DB vs EB. With DB, as you probably know, you do NOT want the finger of your right hand to be perpendicular to the string (as with EB).
Your right arm should be relaxed, yes that DOES include the shoulder, and you should think of it alwasy as a graceful arc. You don't want exactly the same shape as when holding a bow, but same idea. For faster passages, you may find that you need to bring your fingertips up toward the strings a little more and temporarily have that EB-like RH technique, but you'll probably want to drop them back down for most of your playing. Whatever gives you the best tone, do it.
Just keep at it - it will begin to feel more natural, the more you do it. Good luck.
