If you climb into a 55 gallon drum full of water will the weight increase by your exact body weight?

Yes. As long as you don't spill any of the water out of the drum when you get in, the total weight of the drum and water with you in it will be equal to the weight of the drum and the water plus the weight of you, separately. It doesn't matter if you float or sink.

It will weigh about the same as it did before you climbed in. First, the answer above this one is wrong because you cant climb into a full 55 gallon drum without spilling water. As you enter the water, you will displace that volume with your body which weighs close enough to water to weigh near enough the same minus the amount of bouancy.