OK. I don't know how mechanically inclined you are but will see as we go along. Some things may seem too basic for you.
You should have 12 volts at the thick wire on the starter all the time and 12 volts on the thin solenoid wire when key is in the crank position.
A buzzing relay is either a bad relay (swap it with another in the relay box), low voltage to it from the source or a bad ground in the system.
swap relay with another of the same number.
pull out relay and there should be 12 volts on 2 terminals for it in relay box with key in on position. check this.
check all the ground connections and any other connectors you disturbed by swapping engines. Don't forget the ground wire from battery tray to intake surge tank. Make sure no wires or harnesses were chaffed bare or pinched between something.
Try tapping on the relay box to see if buzzing stops.
Electrical problems are hard enough in person but over the internet it is mind boggling. This may take a while so bear with me.
I think you may have missed a ground connection or a connector somewhere while swapping engines. There is also a chance of a pinched harness somewhere. Carefully retrace all of your electrical steps.