I understand Fish and Game is allowing hunters to take two moose in a lifetime instead of one. Under the new rules, if I get my first moose can I apply for a second tag the following year, or do I have to wait?

You have to wait. The new rules allow you to take a second moose of the opposite sex of the one you already harvested. However, as is the case with bighorn sheep, if you draw a tag one year, you can't apply (during the first application period) in the following two years. The exception is second drawings. If there are leftover tags available, you can apply for those leftover tags in consecutive years.