Unless you qualify for a VA loan, you would need to get at least 3.5% down on the $350K house or $12,250 to even be considered for an FHA loan, which has the lowest downpayment requirement. Even then, you should budget to have at least $17-18K to cover closing costs, pre-paid property taxes, pre-paid HOA, and moving expenses. Ideally, you should have $30-40K available and use about $20K to complete a purchase with FHA at that price range. That cash position would leave you with a small financial buffer.
So, if you don't have that amount, you should either consider a condo in the $200K range to be comfortable with your $10K down or save up some more until you have that amount.
Beyond FHA, you'll need excellent credit to qualify for a 10% down with PMI on a piggyback loan. It would be a slightly cheaper option than FHA. In that case, you'll need to bump your cash position to about $45-50K to purchase a $350K condo.