Some lawn mowers such as the Craftsman 37705 have a washout feature. It is essentially a garden hose fitting on top of the circular blade cover. This allows you to pump water into the blade assembly while running the engine. That action is designed to flush dirt and grass debris right out.
Place a drip pan or other container on the side of the mower with the filler. Carefully tilt the mower so oil drains out and into the pan. Discard the spent oil into a proper disposal vessel. Now slowly pour fresh oil into the mower. For the specific type of oil you need check your instruction manual. Make sure to add only as much as your particular model requires. Remember to let the mower sit undisturbed for a few minutes so the oil settles properly inside the engine.
Your Best Digs also recommends the Sun Joe mower but the reviewer also added a number of caveats. He found it will get the job done if you prefer to only bag and just have $100 to spend. However the clippings compartment filled up quickly for him. And the height adjuster felt flimsy and required adjustments on each wheel.