The old Sylvania lights had a plug directly on the light itself, the new ones have a cord with a connector, connected to the plug outside the light well. Do i have to worry about water getting to that?
No, you shouldn't have to. Mine was the same way, and I live in Oregon where it pours down rain almost all year long. Never had any issues.
Also, my passenger side bright light does not work, it didn't work on the original lamps either. That leads me to believe it's a problem with the Jeep & not the light itself. How do I figure out how to make the bright light work?
That's something related to the Jeep itself, not your lights. Since it's only happening to one light, I would almost suspect it's a bad wire connection for that headlight. In this case, a test light would be your best friend.
One more thing, how do I adjust them down? I don't want to blind oncoming traffic.
To aim them, there are two small T-15 torx-head adjustment screws that you can access from the front, between the headlight lense and the chrome bezel. Find the little indentations on the chrome bezels located at the 4 and 10 o'clock positions around the bezel, those give clearance for access to the two adjustment screws. They both aim the headlight so it's a little of both to move it in the direction it needs to go. Top screw for up-down, bottom screw for left-right.
As for how to align them:
How to Align Jeep Vehicle Headlights
When lifting Jeeps the headlights are often over looked so use this simple guide to adjust your headlights aiming pionts.
Step 1 - On level ground, put your Jeep vehicle in front of a wall as close as possible.
Step 2 - With tape, put 2 crosses on the wall that correspond with the vertical and horizontal center lines of both headlights.
Step 3 - Move the Jeep back 25 ft., keeping the Jeep centered with the tape crosses on the wall.
Step 4 - Examine the light pattern.
Low Beams should be hitting the lower right sections of the crosses.
High Beams should split the cross hairs.
Turn the horizontal and vertical adjusting screws (located on the headlights bucket) to make adjustments.