Re: thick distortion
The 'screamo' tone:
High gain tube amp + Les Paul or SG with aftermarket pickups = Fat Distortion.
All those bands have pretty top notch equipment, if I were you I'd start with saving up for a good (tube)amp. If you already have a good amp I'd suggest a guitar without a trem to get a chunkier/fatter tone. I feel that my Ibanez sounds trebly even with duncans in it, it sounds amazing with the JB but is lacking in the oomph/chunk departement.
I don't know what you have in the bridge but I'd suggest a DDn, that one has fat distortion for in the neck position, it's quite hot so you would want something hot in the bridge as well to keep it from overpowering.