Neither - Readied actions come after their triggers
The character Readied an action to draw the weapon and set the trigger to be when breathed upon with a breath weapon by the dragon. Unfortunately for the character, readied actions happen after their triggers.
When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.
Thus, the breath weapon would hit and its effects resolved. Then, the readied action happens and the weapon is drawn. Too late, unfortunately, to stop the breath weapon which is already gone now.
There is no valid trigger that works for this
Any readied action you could set would either take place before the breath attack occurred (doing nothing to it) or after (also doing nothing to it per the above). There is no trigger that you can set that would allow you to (potentially) interrupt the actual attack by drawing your blade during the attack itself.
A fire-breathing dragon breathes actual fire
If you look at red dragons' breath weapon ability - for instance, a red dragon wyrmling's Fire Breath action - it says:
The dragon exhales fire in a 15-foot cone.
Fire and flame are synonymous; thus, if you had some sort of effect that continuously extinguished flame, a dragon's fire breath would also be extinguished if used while the effect was active.