Why have my MOSFET drivers blown in this H-Bridge?
FDD6637 MOSFET datasheet here
TC4428A datasheet here  

Regardless of the survival of the MOSFETs, so far :-), I'd add gate to source zeners to the FETs to clamp Millar coupled voltages from the inductive load.

This may also address your observed problem. Logical analysis suggests it won't :-( - but Murphy and Millar capacitance can work powerful magic. The TC4428 drivers sound nicely robust (if the datasheet is to be believed) with protection against most normal offences. They have an absolute maximum 22V Vdd rating and the ability to absorb up to 500 mA reverse current 'forced' into the output would be expected to clamp inductive feedback via the MOSFET gates. But, gate zeners cost little, definitely help protect the MOSFETs in situations like this, and are very unlikely to make things worse. 



Some power supplies will take no reverse current at all and others do so badly.
Have you checked the supply to see how it behaves? A meter (better an oscilloscope) on the supply during braking may give clues. A very large capacitor may help, but this will help the supply if it is able to dissipate power but not rapidly enough, but only mask the problem if the supply is inherently unable to absorb power.

A resistor in series with a zener (or electrical equivalent) as a load will help braking dissipation (but the zener takes 12/Nths of the power for N volts rise.  

An eg TLV431 switching in a large load as soon as V+ exceeds say 12.5V and dropping it off as soon as order is restored sounds like a simple and low cost solution to absorbing braking energy. 



I have 2 x 300 Watt "wiper motors" (Indian, trucks, for the use of) which I intend to use in a prototype in the immediate future. Should be fun :-).