Batteries are consumable parts and eventually, they wear out. Assuming you have your watch updated to WatchOS 8.x.x, you can check its battery health, by opening the settings app on your Apple Watch and swipe up from the bottom of the page until you see Battery. Tap it and check your Battery Health. If it is close to or below 80%, the battery has used its useful life.
You can pay Apple for an exchange watch (Apple doesn't actually replace batteries in Apple Watch), which will be a refurbished watch of the same model and color for a fee.
Personally, given that the Series 3 watch is now quite old and cannot run the latest WatchOS versions, I would be more inclined to trade it in and use the trade value towards the purchase of a newer, more modern Apple Watch.