I don't think I can find one at around 10k
Man, these restrictions suck. Any car that's as new as or newer than your Versa is going to be either more expensive than $10k, or no more reliable than the Versa. A well-designed older car (like, say, a 2011 Corolla) CAN be more reliable, but you're going to be spending money fixing wear items more often if you want to keep it that way.
Speaking of which, are there any geographical concerns or issues with local dealerships/mechanics that are a problem? My grandparents aren't huge Ford fans, but they're on their fourth one, mostly because they live out in the sticks and have to drive three towns over to buy anything else. (They'd still be on their second if those cretins at the dealership could fix anything that wasn't an F-series pickup. But I guess a three year old Hybrid with a busted computer is basically unfixable, right? Having to have it fixed multiple times and then totaling it out would be totally normal. Dumbasses.)
All that bitching aside, I'd take a good look at a new (not used) Mitsubishi Mirage. It's got a CVT, and is likely to have similar problems to the Nissan in the reliability department, but a 10 year powertrain warranty (if you're the original owner) is a nice counterbalance to that.
A quick search shows a couple new ones near me listed for $12k, with oodles more for $13k, so that might be a stretch goal, or maybe something you can get a deal on if you corner a desperate salesperson towards the end of the month. (You could talk to your bank too - often you can get better financing or lower interest rates for a new car than for a used one, which will offset some of the additional costs.)