Actually, your drop down list is likely to have two issues, you've described one, the other is that it won't auto-complete when the first field is the ID number field.
The issue you describe, showing the the value of the second field after you exit can be easily remedied. Assuming you have a relationship to the field of values that matches by the ID number, you can add the second field from the related table to your layout. You can even use field behavior settings to deny access to the field, give it an opaque fill color and place it on top of the drop down list formatted field. When you click or tab into the field, the drop down list hidden behind the field pops to the front and deploys. When you then exit the field, it disappears back behind the 2nd field that then correctly displays the associated text for the ID number that you selected.
Setting up the field so that it auto-completes, however, is something that requires scripting and some other "under the hood" fiddling to make it happen.
See this demo file:
Thanks so much!
Another question: When I use a drop-down list and display a foreign field over the other field as you explained, after hitting "enter" to select the drop down, I have to hit tab twice to get to the next field. How can I go directly to the next field after hitting enter? (I have the field set to go to the next object upon hitting "enter".)
I'm afraid that may be just the "nature of the beast".
Might be worth mentioning that the first problem Steve cited there, where the aim is for the user to be able to type a few letters and hit return and select from the value list without using the mouse, can be done using a pop-up menu if you're on a Mac - just not on Windows. Presumably the OS does not allow it.