Wow! You are in the driver's seat. I'm no mortgage expert and certainly no authority on what's going on in Connecticut, but here's the little I know.
The higher the downpayment, the lower the amount you are going to finance. So, if you want a $500,000 home but want payments more in line with a $250,000 home, then you would pay 50% downpayment.
The more points you pay, the lower your interest rate will be. A point is 1% of the loan amount. As a general rule, each 1% point you pay will lower the interest rate about 0.25%. So, if you paid 3 points, your interest rate would go down 0.75%, 4 points = 1.0%, 5 points = 1.25%, etc. In other words, a 7.5% interest rate, with 2 points paid, would be 7.0%; with 3 points paid, 6.25%. You should check with several lenders in your area to find out what the interest rates are in your area.
Here's my (maybe wholly unqualified advice): you want a mix of four things:
1. high downpayment to lower the loan amount
2. paying a couple of points (or more) to lower the interest rate
3 keeping some of the money to pay off consumer debt (credit cards, etc.)
4. keeping some of the money in savings to grow
So, if you buy this house for $515,000, put 38% down (or about $200,000), the loan amount would be $315,000, and each point would be about 3,150; pay 4 points (or about $12,600) to reduce your interest rate a full 1% (or more depending upon rates in your area); that would leave you with $37,400 to pay off high interest consumer debt (if any) and the rest to put in liquid savings (CD's, savings accounts, etc.). Or any combination of the above.
You are in great shape! Congratulations.