Don't assume the WiFi signal strength you're seeing on your phone matches what the thermostat experiences - they can be very different, and a device constantly hunting for a stable connection will burn through batteries in weeks. Before investigating the C wire, try moving the thermostat closer to your router for a few days as a test; if the drain noticeably slows, poor connectivity was likely your problem all along. If it doesn't improve, then yes, the missing C wire is almost certainly the issue, and you'll need either to run one from your HVAC system or install a power adapter accessory that Nest makes for exactly this situation.