A relay merely relays power. It shouldn't be possible for a relay to burn up w/o a fuse in front of it being the wrong size.
Start off by double checking the sizes of all the fuses according to the fuse panel cover.
Is your new box the same year/model as the car?
On the buzzing, I've seen this in two cases. One if the relay is bad (which doesn't happen often) and two when the power/ground being supplied to engage the relay is not the nearly 12volts that it takes to engage the relay.
More info about relays can be found here the12volt.com under the basics section. That should help you understand them better if you are unsure about how they work and why they work etc. I'd use that information to test the actuating power/ground to ensure it's good. You may find that a ground is bad and not allowing the relay to fully actuate. Thinking about this further...a poor ground could have made a bad relayed connection and caused the stuff to burn up. This would likely be a ground before the pcm.
Check the grounds at the pcm, behind the battery and on the post of the transmission by disassembling, cleaning and reconnecting. If you get that same buzzing, I'd check the wire from the pcm to fuse box.
