Hi Truffade,
We did this 2 years ago, bought a new house before selling our old one, our new house required a lot of work, so buying and selling on the same day was not an attractive option.
We did not clear the mortgage on our old house until the house was sold, about 11 months after we bought the new house.
There is no requirement to pay off the mortgage on the old house, when drawing down a new loan. Approach some banks and see what they can do for you.
With your comfortable income you should be ok on borrowing capacity, however your downpayment is small. Banks are looking for 20% down for second time buyers, anything less requires an exception to the central bank rules. I'd imagine they would make an exception in your case, but currently you have less than 10% deposit, which might be a problem for them. Keep on saving and don't make any more lump sum payments on your existing loan for now.
I'm not sure that involving your relative here will improve things for you when looking for a mortgage, you're simply replacing one loan with another, the new one, while lower interest, has a shorter term and higher repayments. If anything this may reduce the amount that a bank would lend.