First of all congratulations!
The main item to concern yourself with is the tax consequences of the $350,000. If the place where you won already deducted for taxes then you are OK but you still are going to have to report it as income and then deduct the pre-paid taxes to ofset it.
If you sell the home then the taxes will be due on your next tax return. The difference between what you originally paid for the property(in this case $0) and what you sell it for less selling expenses.
Once that is handled then there are NO taxes you have to worry about when paying down your loan. You will be losing a good tax write off though which is the deductions that you get for owning a home that has a mortgage and write offs that you get for the loan costs (ie:interest).
If you are thinking that you can sell the new house and offset that tax liability by putting that money into your existing house – it doesn’t work that way.
To do a tax deferred exchange (1031) you must sell and then BUY within 2 years after that date. You can see that a property that you already own doesn’t qualify.
If you would like some assitance regarding this I would strongly recommend that you consult a tax consultant.