12 volt relay shorting out?
is it possible that charging a 12 volt system with a 24 volt system [does something strange]?

Stop and actually think about it.  You are really asking "If I abuse something, could it then possibly no longer function as specified?"  Seriously!?

You said yourself that the two 12 V car batteries in your system are wired in parallel.  They therefore together act as a big 12 V battery.  You tried to apply 24 V to two 12 V car batteries.  If this 24 V charger was able to supply sufficient current, then those batteries are now junk.  You are lucky that the resulting gasses didn't explode.

Whatever electronics is connected to these 12 V batteries probably wasn't designed to handle 24 V for very long.  It could also be dead now.  In fact, that sounds like what happened.  The over-voltage caused something to blow, which permanently turned on one of the relays.

Your mention of a "1080p" makes no sense whatsoever in the context of a dump bed going up and down.  Without further definition (note that you didn't supply any), "1080p" means "1080 scan lines, progressive".  That's common for HDTV, for example.  1080 is the number of scan lines on the screen, and "progressive" means all are drawn each refresh period.  In this case, "progressive" is the opposite of "interlaced".

Next time, pay attention to the voltage and other ratings.  Or, get someone who knows what they are doing.  Connecting 24 V to a 12 V system is just plain dumb.