I've had this happen two times now on my 70hp Johnson, Camo has has it happen on his boat too (mariner)...it was the relay every single time...cheap, easy fix too. Most motors use two relays, one for up, one for down...The relays are two black cubes next to each other about 1" in size. They should be in an obvious spot on the motor. To test it, just switch the relays (they pull right out). I think they are $12 each at NAPA.
yes, It totally sounds like the solenoid/replay. There were three on the side of my engine, on the driver side right in the middle. It is a very easy fix. Find the bad one and take it with you to the marina or napa.
My bad one is now fixed too, so I keep it in the boat as an extra. Inside this little box are 2 fine wires that are soldered and one became undo and wasn't making contact. So my grandfather-inlaw soldered it up and put the back plate back.
A very easy fix, thanks to BigRed, after pulling my hair out, trying to find out what was wrong.
