Take a look at all of your fuses to ensure they are good <under dash and under hood>.
I would tackle each item separately.....and meticulously go to the next.
As far as draw on the battery, I think I'd give the new battery a chance to perform.

 

Like jtpeters said, tackle these things one at a time. Yeah, test all those fuses!

1. Fuses. Check them all! http://www.wjjeeps.com/fuses.htm
2. Wipers. Relay 32 & 33, breaker C1. If it were me, if the relays and circuit breaker are good, I would find the wiper motor and test the connector pins to see if it's receiving power when you turn them on.
3. Headlights, low beams. Fuses 14 & 15, 10 amp. If the fuses are good, maybe the bulbs are bad. Pull the headlights and check them. http://www.wjjeeps.com/lamps/lampsrem.htm#EXTERIORREM
4. Windows. Fuse 12, 50 amp. If the fuse is good, maybe you have a broken wire in the boot, a common problem. https://www.jeepforum.com/forum/f31...com/forum/f310/2002-bella-blue-wj-build-thread-4239282/index4.html#post40623861
5. Blower. Fuse 1, 40 amp. Hopefully it's just a fuse. If not, there are people here who have had to deal with this.

 

If it is 4.0 there is the main wiring loom that goes behind the head that likes to rub the head bolts. Check if that has worn through the insulation. If it has you can pigtail it together and put protection on.

 

Check to see if the low beam bulbs are burned out. I have seen cars with no low beams because the owner didn't even realize it when the first one had burned out, so when the second one burned out they were certain that there had to be a serious electrical problem. :surprise:

 

Some good info from others. It's not unusual for rodents to move into any vehicle which has been sitting for a while. In checking for chewed wires up under the dash also check the wiring harness in corners under the hood in front of the cowl area.

Also not a bad idea to remove the under-hood cowl seal, remove just a few fasteners, and you can lift up the plastic cowl cover a little w/o taking the wipers off to look for signs of rodents/mice. You can see how to do this on your 2004 in the below write-up.

https://www.jeepforum.com/forum/f310/cabin-air-filter-housing-write-up-3717505/

I ended up with a mouse nest behind the windshield wiper motor/linkage on my ZJ which can be seen below. Not pretty but luckily that was just water which had soaked the nest.

 

Concur with the others on a detailed check for critter (rodent) damage. I bought a WJ that had been sitting under some trees for 18-24 months before it went to Goodwill auction and I picked it up. In my case, the speedometer, horn, wipers, and AC didn't work. Also, the blower fan made an odd sound that I initially thought was a recirculation door that had broken off and was hitting the fan. I found a large cache of acorns in the airbox, so started checking the wiring. Ended up finding 5 wires with sections chewed out of them in the harness coming from the firewall along the driver's fender well. After soldering in new wiring to replace the damage, my electrical issues (along with several fault codes) disappeared. I pulled the glovebox out to check the recirculation door and found it intact. After dropping the blower motor though, I found a large mouse nest in it and many of the blades, and even the base of the "squirrel cage" gnawed on. Replacing the motor for $5 from the salvage yard fixed that problem just fine. In my case, the chewed wires in the harness weren't all in one spot. They were throughout the harness. I actually used a magnifying glass and bright light to check along the harness. Some of the wires were severed flush with taped portions of the harness and very hard to see, with a several inch section chewed off to flush with another taped section. Checking all exposed wiring throughout a vehicle that has been sitting for a while can be time consuming, but pay huge dividends.