You are looking for a simple answer, but your question contains many variables so my answer will be a complex. First, let us look at a constant in the equation: 35%. That is the ratio banks look for as the maximum percentage of debt a household should have as a percentage of income.
Let us compare you, with zero debt, and a neighbor, who pays $500 per month in credit card debt. Let us say your neighbor earns $500 more per month than you do. Assuming equal down-payments, you can handle a larger mortgage than your neighbor because your debt ratio is lower. If you keep your consumer debt in check, you will probably have a better credit score, which will merit a more attractive rate, all other things being equal.
Now let us add the down-payment variable. A 20% down payment reduces your principal as compared to a no-down or 3.5% down payment. It also buys you a lower interest rate, makes it easier to qualify for a loan, and removes the requirement to buy private mortgage insurance. How much a 20% down payment gets you in terms of more attractive rates depends on your credit score, the markets, the area you are buying into, and other facts. You need to weigh the benefits of waiting to buy a house with a 20% down payment verses buying sooner with a smaller down payment, and can do so now with a little shopping online. (Get a free mortgage quote to see where you stand now.)
Regarding your question, the difference between 3.5% down and $5,000 existing debt verses 10% down and no debt is huge. Again, the rule of thumb is that your debt load should be no greater than 35% of your monthly household income. For the sake of argument, let us say your household income is $4,000 per month. That means you can afford a $1,400 debt load. If you pay $300 each month to pay down the existing debt, that means the bank will figure you can afford a $1,100 mortgage payment. On the other hand, if you have zero credit card debt, the bank will figure you can afford a $1,400 monthly payment.
What is the difference between a $1,100 per month house and a $1,400 per month house? Assuming a 5.5% interest on 30-year loans and $2,000 in annual property taxes, a $1,100 payment buys a $165,000 mortgage, and $1,400 payment buys a $230,000 mortgage. Granted, the more expensive house will have higher property taxes, so this illustration is somewhat exaggerated. I am trying to make the point that driving down your other debts frees you for a larger mortgage and a pricier house in a nicer neighborhood.
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In conclusion, it's not a matter of credit card debt (less is better), or a bigger down payment (bigger is better), the real question is how much of your mortgage you can squeeze into your 35% household debt load.
I hope this information helps you Find. Learn & Save.