Did Gene Watson record some of Conway songs?

Yes, Gene Watson has covered at least two songs that were made famous by Conway Twitty.

1) "Hello Darlin'" (original released 1970 by Conway Twitty) was covered by Gene Watson on his 1975 album "Gene Watson Sings Some Conway Twitty".

2) "Crazy In Love" (original released 1983 by Conway Twitty and Loretta Lynn) was covered by Gene Watson in 1986 on his album "Reflections".