The main differences between a combo and a big bands are, for the most part, the number of members, amount of improvisation involved, and types of charts played. Usually, a nonet is the largest “accepted” combo, although that’s debatable! I’ve heard the term ten-tet thrown around a bit, as well. Charles Mingus’ bands and the 1969 Ellington White House tribute band are great examples of how a large combo/small big band can sound. With the size of my school I’ve never had a full big band – always short of trbs – so what you’re describing is the reality I live with daily!
With eleven you’re right on that cusp, and your classification really depends on the rules of the festival you’re planning on attending. In terms of the band’s sound, one option would be to to see what instruments might be doubling notes in the voicing, i.e. 2nd tenor/2nd trombone, etc. and make sure you’re covering the voicing. Rewriting parts is another option there. Also, Hal Leonard has a great Small Big Band series for bands such as yours.
Usually, a combo involves playing a less arranged chart – more of a head chart generally. Typically, you’ll have more students improvise in combo due to the lack of needing to learn section solis, etc. thus allowing you more time to focus on improv. What rhythm instruments are you missing?