So I waited a bit to give my 2 cents
First how did you know the FOH mix sounded bad ( IE were you out front of the speakers during your set ? )
It should only take a minute it dial in something EQ/level wise FOH , I have also had plenty of "musicians" that have NO IDEA the sound out front is FAR different than what they were hearing onstage ( not meaning you as I don't know you
)
There should be no issues with a decent PA to get your guitar sound out most PA systems have far more power and a better frequency response than any guitar cab.
As far as your sound on the stage THAT can be an issue Most floor/wedge monitors are Very flat sounding EQ wise. I used to use a 1x12 cab next to my floor monitor so I could get my sound back to me with only vocals and bass in the monitors
I have had my best luck live sound wise with the mic ( SM57 ) 6-8 " away from the cone at a 45 degree angle to the side of the speaker. or using a direct box out of the extension out of my amp.
Happy Jammin !!