The Uzbekistan [e-visa portal][1] confirms that US citizens over the age of 55 are visa-exempt for tourism not more than 30 days long. The page says:

>  For citizens of China (as part of tourist groups), the United States and Vietnam who have reached the age of 55, a visa-free regime has been established for entering the Republic of Uzbekistan for tourism purposes for a period not exceeding 30 days.

It appears, then, then all you need to enter Uzbekistean is your valid US passport, and that no visa is necessary. 

Remember that if your travel involves layovers or stops or transits in intermediate destinations, you will have to be admissible into those places or able to transfer there to your onward flights. That will, in turn, be dependent upon the specific routing and airport design of the intermediate points.

  [1]: https://e-visa.gov.uz/application