The data came from Retrosheet. The graph below shows the % faced in the AL.
Now for the NL.
Now for both leagues in the same graph. The AL is the red line and the NL is the blue line.
This last graph shows the difference between the two leagues (NL - AL). In the first year, 1953, the NL had 0.288 while the AL had 0.259 for a difference of about .029. Then the next year the NL was .04 higher. It is intersting to see that there was one trend to about 1970 of the NL edge falling (actually turning negative in 1960 and staying there until 1970, except for 1962). Then there is a trend for at least 10 years of the NL rising relative to the AL. Then it generally declines until about 2000 and then it starts rising again. Maybe the DH plays some role here but it can't explain all of it.