West Coast,
I admire your attitude and work ethic. You deserve to do well and I am sure that you will.
I would keep the job centre informed as to what you are doing. You clearly are a man of integrity, which will stand you in good stead with the j.c. Honesty is not the best policy, it is the only policy!
Keep a record of what they tell you, in writing if possible, so that you can cover your back in the event of them giving you any incorrect advice (quite likely).
Very best wishes,
John.