The Dodgers would not be able to move to LA without another team moving west, and no other NL team would be interested. Depending on how soon the Dodgers want to leave NY, they could move somewhere else or wait until the first wave of expansion and move west alongside an expansion team in the West.
Obviously the Senators would not have moved to Minnesota, they would try to move to either SF or LA instead. The Senators would not be able to move until the first AL expansion, when OTL they moved to MN and DC got an expansion team.
Without the Dodgers and Giants moving to CA, the Continental League would have had different membership but probably a similar effect (replace the NYC and Minnesota teams with SF and LA). Both leagues would need to expand.
I would suggest this timeline:
1956 - Giants move to Minnesota
1960 - Dodgers move to LA. The NL expands to include the San Francisco Seals and Houston Colts.
1961 - Senators move to LA and become the Los Angeles Angels. The AL expands to include the Washington Senators and the Atlanta Crackers (this was the name of the minor league team at the time, and besides it's no more racist than the Braves).
After this the next round of NL expansion will be moved up to include a replacement for the Dodgers, and after that I expect that which markets are filled by MLB teams will more or less be the same as OTL.
Which teams are historically strong/weak will probably be the same - big-market teams like the Yankees / Red Sox / NYC NL team will have lots of money to spend, and the MN Giants and the Pirates will not. Which player is on which team and which team is good in any given year will all be changed by butterflies.