I'd say check all fuses and relays starting at the main box under the hood. If electricity is getting to minimal things and not others on top of getting weak current to the things that are working, I would imagine a possible bad ground or fuse/relay malfunction somewhere, considering the batery is brand new. When I had my Wrangler, there was a paper thin (looked like tissue paper when I cleaned it off) build-up of corrosion on the negative battery post, and that prevented the whole vehicle from getting enough juice to crank. I'm always surprised when the smallest things in an electrical system can cause problems. Just my
