How to quickly cool a test tube without cracking it?
Rather than working on your technique, check whether other test tubes are available.

Apparently, a glass-like material with a very low thermal expansion coefficent would be a good choice.

The glassblowing shop at the University of Delaware provides a nice overview of the possible materials and their properties, and there are definitely additional online resources, e.g. by manufacturers like Schott, Corning, etc.

Here's a summary (coefficients are given in $cm/cm/°C$):

soda lime glass $89 \times 10^{-7}$

Pyrex, Duran $32.5 \times 10^{-7}$

Vycor  $7.5 \times 10^{-7}$

fused quarz $5.5 \times 10^{-7}$

Please note that quarz is both pretty expensive and demands special care! Alkaline solutions and quarz are a bad conbination!