If what you're recording doesn't sound like what the guitar normally puts out, yes it will sound wrong when reamped.
Seems like you're overdriving the DI. If it's active DI that could be due to clipping, if passive DI...um...transformer saturation?
It sounds worse with EMGs presumably 'cos they have higher output and so overdrive sooner, and perhaps also 'cos if you're running them off 9 Volts they clip themselves anyway, so you'll be distorting a signal which is already clipped. Guy called Mutant posted some pics showing that on this forum, a couple years ago I think...
Does it stop happening if you back off the guitar volume? If so the solution would be a DI box with greater headroom. (Backing off the guitar 's volume pot is not a recommended work-around for recording as it alters the tone as well as the level from the instrument.)
Oh, or possibly it's clipping at the DAW's input, not the DI... you can clip a -10dBv line input with raw guitar, don't let anyone tell it's impossible. "Instrument level" is really just a kind of nominal minimum.
