Ask for written confirmation of the verbal offer. The written confirmation has to include the position title you are going to have, the start date, your new location and the name and contact of whoever you are reporting to.
In the meantime, continue with the interview process for the second position. Abort the process the minute you get written confirmation for the first position. Since you already did the homework for the second position, you might as well hand in that homework. Simply let them know that you are aborting the process shortly after you get the feedback for the work that you handed in.
Don't say anything to HR until events have developed, i.e. you have a written confirmation letter in hand about the verbal offer, to the point where you can give a straightforward narrative to give to HR - that should take maybe a couple of days. Say anything to HR while your status is still unclear and events are still in flux and you will have nothing but a frustratingly convoluted narrative to give to HR. So just wait for the couple of days until you have your ducks lined up and then initiate contact with HR and get your communication over with.