According to this, the number one country song in the nation the week I was born was "I Wanna Live" by Glen Campbell.  (I don't recall ever hearing it, but I looked up the lyrics. "Rover is big, Tabby is small"? Good grief!)

At least the number one song on Billboard's Hot 100 that week, "Tighten Up" by Archie Bell & The Drells, wasn't bad.

EDIT 10:21PM PDT: After I posted this entry, I thought about the fact that Billboard gathered the survey data for its lists from one week and then declared the resulting lists were for the following week (ending on a Saturday) [correction in following edit]. So, to be a stickler about it, the singles that were most played and bought the week that I was born were named the number ones for the week ending May 25, 1968. Therefore, during the week that I was born, the record that was in the process of becoming the number one record on the Hot Country Singles chart was "Honey" by Bobby Goldsboro. Wikipedia reports, "The Cincinnati Inquirer reported that the song frequently appears on 'worst songs of all-time' lists, and in April 2006 CNN named it the 'Worst Song of All Time'."  Fabulous. Over on the Hot 100 chart, "Tighten Up" was on its way to being number one again. However, I see that Simon & Garfunkel's "Mrs. Robinson" was number one the following week (i.e., for the week ending June 1, 1968). Man, it would've been awesome if that record was in the process of becoming the number one record on the Hot 100 chart during the week that I was born.

EDIT 11:04PM PDT: AAARRRGGHH! And YAAAYYY! According to this, if I'm looking for the record that was in the process of becoming number one on the Hot 100 during the week of my birth, I have to look at the record that was named number one for the week ending June 1, 1968. So "Mrs. Robinson" it is!  I can't find a similar description of the compilation of the Hot Country Singles chart, but as far as I can tell, "Honey" was number one on that chart for the week ending June 1, too (and the week ending June 8), so no change there.