A key point: Dewey will not appoint Vice President Warren to the Supreme Court, of course, but he will name two Justices in 1949 (assuming Frank Murphy and Wiley Rutledge both die, as OTL). These nominations will have to be approved by a Democrat-majority Senate. That should not be a serious problem, as Dewey would be very unlikely to nominate anyone so controversial as to provoke a bloc rejection by the Democrats, or indeed any serious deviation from the tradition of near-total deference to Presidential choices. (It should also be noted that SCotUS picks were much less contentious in that era: in 1945, Truman appointed Republican Senator Harold Burton to the Court; Burton was unanimously approved the day after Truman nominated him.)
Truman's 1949 appointments, Clark and Minton, served until 1967 and 1956.
As to Congress: suppose the PoD is that some Republican leadership figure, Frobble, objects to Dewey's play-it-cool strategy, and insists that Republicans make an all-out get-out-the-vote effort. When somebody objects that this might stir up additional Democrats to vote, Frobble says "We outnumber them - if you don't believe that, then you're arguing that we are trying to sneak into the Presidency against the majority of Americans." The RNC and Dewey's campaign managers agree, and launch that GOtV effort.
My analysis strongly suggests that the 1948 popular vote was about 7.5M below the historical trend line, as indicated by the 1932, 1936, 1940, 1952, 1956, and 1960 results. Suppose this deficiency was largely due, as many have thought, to complacent Republicans staying home because "the election was in the bag". Some of it was despairing Democrats staying home, but AIUI, the Democrats did push GOtV, minimizing that effect.
So the GOP GOtV turns out an additional 7M votes, all across the country. That's enough to give Dewey a solid popular vote victory (28M to 24M) and to flip 16 states with 167 electoral votes, for a 356-136-39 victory. It will also have downticket effects. A quick survey suggests that it would flip about 50 House seats, which would continue Republican control. But it would only flip five Senate seats, so the Democrats would still gain control there. (This would be the first election to produce a Republican House and Democrat Senate. OTL, that didn't happen until 2010.)
Thus Dewey would have to govern with that constraint.