Leds for exterior use that jut attach to the ceiling but you will need electrical run to that place mybe a simple conduit painted to almost disappear .

Outdoor ceiling fans are great to have for a covered patio.

"I do not want the added expense to use tongue and groove to cover it"

Does this mean that visible cables/conduit and electrical boxes attached to the ceiling are acceptable to you? Will it also be acceptable for the same items to be visible on wall(s) for the switches that will control the ceiling lights?

Yes, I was hoping for options with least visibility of boxes/cables. The other option I was debating is boxes on the posts that are next to house and just doing string lights.

String lights are a nice option.

I'd follow dark sky principles for any outdoor lighting which means lighting only the areas that need it (i.e., not into the yard, windows, or onto walls), not overlighting, and using low-color-temperature lights.

I like the cylinder lights like those in this photo:

• Progress Lighting Cylinder Collection 6-inch Black Modern Aluminum Outdoor Ceiling Light
• Access Lighting 1-Light Black LED Outdoor Flush Mount Light

Yes, it would require conduit and wiring but, as suggested above, the conduit could be painted to make it less obvious as shown in the photo.

Thanks!