Does buying a less expensive home help lower your mortgage interest rate?

My husband and I have been pre-approved for for a decently large mortgage loan amount but our interest rate is not quite as low as we would like. Does buying a home that is much lower than your pre-approval amount help lower your interest rate? Also when we were pre-approved we were about $ 3,000 in debt. If we cut that in half will that have any real noticeable affect on interest rates?

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