I have about 100 cities all over the USA and want to calculate some distances. According to that I want use one coordinate projection and need to know which one is the best so I can get the most accurate results. I have compared the results using NAD83 for Texas and NAD 83 for Washington for a same data and the results are not the same.
If you want to calculate distances then you should use an Equidistant projection that best covers the USA. USA Contiguous Equidistant Conic would likely be an appropriate choice.
Your best workflow may depend on whether you want "as the crow flies" distances or "via road" distances. For the latter, you would need to have a line data layer. For the "as the crow flies", you should look at tools that support a "geodesic" option. If it supports a geodesic option, that will give the shortest distances on the ellipsoid surface. The calculations won't take elevation into account though.
As Vince points out in comments, equidistant projections have very limited equidistant lines. For a conic or cylindrical equidistant projection, the standard parallels and all meridians only are equidistant.
A simpler option might be not to use a projection and compute distances directly from geographical coordinates using the Haversine formula.