I believe the technique is to duck under the barriers in front of the team, avoid getting shot (duck if necessary), then politely ask the crew chief for each aircraft to open the little panel, under which is the aircraft serial number. Not easy, but it has been done in the past. Note however that the team number on sides of air-intake is not permanent so may be switched between aircraft between performances. So you cannot assume that aircraft "1" (for example) on a particular day is the same aircraft "1" on another day or even later on the same day. Also, there are often two F-16 two-seaters coded "7" present with the rest of the team. Given the obvious difficulties, it seems unlikely that you will ever be able to positively identify the Thunderbirds F-16s you saw on 21st May. You could of course contact the team (by email is best) and ask for the tail numbers of the six aircraft that were at Shaw on the 21st - this could work and I (and others!) would be very interested to know if it does.
Best wishes for Christmas, TM74