Can you refinance a interest only construction loan prior to converting to conventional?

My wife and I recently moved into a new home which we are still paying interest only on the loan. We also just sold our other home and would like to put the money we made off of it down on the loan. Can we put the money down when we convert or do we have to refinance?

Should I REFINANCE a construction loan to a permanent loan OR convert to a permanent loan?

It seems that interest rates are much lower for refinancing existing loans, as opposed to new loans. I have a construction loan which doesn’t expire for a few months, would it be better to refinance the construction loan to a permanent mortgage or do a construction to permanent conversion loan? Which will get me a lower interest rate?