Geography also plays a part.

I know, with the current Washington Nationals, the city had to wait until a National League opportunity arose (even though DC had always historically been associated in the 20th Century with the American League). No way was the Baltimore Orioles owner going to let another AL team compete with the Orioles from only 35 miles away. This, despite the Griffith family magnanimously allowing the Browns to move to Baltimore and encroach on their "territory" in 1954.

I think that problem started when The AL expanded by adding 2 more teams, but The NL didn't follow suit.  It was awkward having one league bigger than the other.  That hadn't existed since the 1880s .

I guessed both Gomez and Power a few clues ago. Gomez came back out of nowhere as a relief pitcher with the Twins at the age of 40+ or so. Power was known for the then sacrilegious habit of catching throws one-handed! 

The current Milwaukee Brewers of the NL were traded from the AL a few years back. I can't say why, but I don't like it when teams change leagues.

Yes, I remember that the "interesting tidbit about the player" text blurb said "Vic once stole home twice in one game!", and had a picture of a Black runner sliding safely into home.  I think it was the Topps 1959 records card (1960 season release). I had always seen it spelled "Pellet".  I wondered why he had such a strange middle name.

1. Gomez was the SF Giant starter. He pitched a 6 hit, complete game shutout in the Giants 8-0 win. I remember listening to that game (as I bet many NY area fans did). Sort of a sad but ineveitable day, looking back.

2. Vic Power is indeed the guy. He wasn't a big 1B, as first basemen go. Under 6 feet. Guess he could run a bit. Btw, "Power" was an alias. His real surname was Pellot. Both steals of home were very late in the game against the Tigers -- first off Bill Fischer, the second off Frank Lary. Charlie Lau was the catcher.

1. Johnny Antonelli?   I'd have picked him over Ruben Gomez.

2. Vic Power!!!  How could I forget?

1. Drysdale is correct. Marichal is not (he didn't come up to the Giants until 1960).

2. Nope, not Cepeda

8. Billy Hoeft is correct

So, two players left -- the SF Giants starter in that historic West Coast opener, and our home plate stealer (2x in the one game). 

1.  Juan Marichal and Don Drysdale?

2. Orlando Cepeda?

8. Billy Hoeft?

I haven't even heard a word about Baseball since the 1870s, other than hearing that The Cubs finally won another pennant since 1945, and won The World Series - I remember that occurred sometime after 2000, but no idea what year.  I was also told that The Houston Astros moved from The National League to The American League, but have no idea when that happened. That's the only instance I can remember of a team changing leagues both in The Majors, since 4 teams from the Defunct American Association joined The National League in 1892.  So, I guess that unless The Astros have won The American League pennant, no team has done what you asked with both championships occurring after 1900. The St. Louis Browns of The American Association transferred to The National League in 1892.  And after a team name change (same franchise, but new owners), won The National League pennant in 1926 and many years after, and went on to win several World Series.  I seem to remember that The Old Browns of The Old Major League AA  won a  "World Championship Series" vs.  The Chicago White Stockings of The NL, in 1885 and/or 1886.