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You can try that, but it isn't likely to yield results. It's like watering potted plants without putting the seeds in. Most of the tossed grass seeds sit on top of the soil or existing grass blades and never germinate. The seeds need to make it into the soil, just like growing a potted plant. The best way to accomplish this is to aerate your lawn first, then overseed.

There are several possible reasons why your Bermuda grass is not thick, with the biggest reasons being too little water, too much shade, compacted soil, drainage issues, and too few nutrients in the soil. Bermuda grass needs frequent and direct sunlight to stay thick, so thinning grass is likely the result of a lack of sun. 

You may need to lay sod on the areas you dug up once you get the okay from the contractor. It will take about 30 to 45 days for the sod to take root. Consult a local sod installer to get an accurate estimate of the timeline of your specific yard-enhancement project.

To prepare your lawn for core aeration, water it the day before you aerate. (Alternatively, you can wait until it rains and aerate the following day.) Aerating after watering or a rainy day will make the process easier since you can penetrate moist soil better than dry soil. You should also flag your sprinklers so that you don’t accidentally hit them while aerating. If you have a lot of thatch buildup in your yard, you might need to dethatch your lawn before aerating as well.