Astra had twice been referred to as "the 6th princess of the 6th dynasty of the 6th royal house of Atrios" but when the Shadow tried to give that line later, he messed it up, calling her "the 6th child of the 6th generation of the 6th dynasty of Atrios".
Astra started to glow and transform into the segment of the Key to Time before she touched the tracer (in fact, she never actually touched it) all of the other segments didn't transform until after the tracer had been in contact with the disguised segment for a couple seconds.
Romana expressed surprised that she was meeting the White Guardian instead of the President, even though the Doctor had previously told her that she had actually met the White Guardian in disguise as the President, and she herself had referred to both Guardians.
When Drax shoots himself with the dimensional gun, it shrinks him to a fraction of his original size, but the effects show him growing instead of shrinking.
The concept that Astra is the 6th princess of the 6th dynasty of the 6th royal house of Atrios and is therefore somehow linked to the 6th segment of the Key to Time doesn't wash, because the Doctor and Romana could have searched for the segments of the Key to Time in any order. If they had sought this segment third (for instance), then the whole chain of "logic" wouldn't work..
When Drax is sitting next to K9 inside the TARDIS, the shadow of a boom microphone can be seen on the wall above his head.