Ok so I've been told that it is possible to use the charge and exploit weakness powers simultaneously but that it is not possible to do the same with running shot and psychic blast, but reading the power rule book it would seem like if you could do one then you could do the other, what is the actual rule for both?

Both of those examples work together just fine.
Charge works with exploit weakness, Running shot works with psychic blast.

Just remember that you cant base your opponent during running shot.
Psychic blast also does not work when based.

These powers do work when based if the character in use is a sharp shooter.
So far there are only 2 sharp shooters in the game.
Hitman and Star Lord.
Star Lord is the only sharp shooter with running shot, Hitman is the only one with psychic blast.