All the what-if's regards to Dewey becoming President having him winning in 1948 in his second attempt at the Presidency. It is often forgoten that he ran in 1944, when he went down to a 432 to 99 defeat in the electoral college, but that doesn't show that the race was acutally quite close espically in the large states. Dewey lost Michigan (19 votes), and New Jersey (16 votes) by just over 1%, and went down to very narrow defeats in Pennsyvania (35 votes), Missouri (15 votes), Illinois (28 votes), Idaho (4 votes), Maryland (8 votes) and New York (47 votes) (by just 5.01 %) in his home state.
Dewey wins these eight states, he wins the election by 271 to 260 elec votes.
How does Dewey deal with the end of the war, and the Soviet Union in Europe?
First, I'd take issue with the opinion that the above states are swingable: a 5% margin in New York is a very big margin. Furthermore, Dewey is running against FDR as the Americans are winning. For my money, you'd need a more substantial POD than simply some handwavium on election night. The one that requires the least outside intereference is that FDR is sick in the fall of '44 and unable to campaign vigorously (and hence seeminly disprove any rumors of his ill health).
To the actual question, as to the conduct of WWII there are really two key points: the Yalta (and Potsdam) Conference(s) and the use of the Atomic Bomb against Japan. I have a hard time seeing the Conferences yeilding a substantially different result, since Stalin seems to have presented the Churchill and FDR with a list of requirements. Dewey may decide not to press for Soviet involvement against Japan, but that would be a big gamble, depending on Dewey's knowledge of the A-bomb. As to the latter, I'd have to imagine that if presented with the option of potentially ending the war without an invasion, Dewey would have taken it, as Truman had. There might be some question, though, as to whether the change of administration might force some delay in the development of the bomb. I doubt this, since I imagine Dewey will need to take great pains to make sure that as little as possible about the running of the war is undisrupted. A shrewd move in this regard is to keep most of Roosevelt's cabinet (at least the Sec War and perhaps the Sec State) until the end of hosititilies.
While Dewey was certainly an "interventionist" in 1948 and the context of the Cold War, his campaign in '44 was centered on domestic policy and against the New Deal. If he takes an aggressive stance on that front he may find himself with too great a distraction to take a strong stance against the Soviet Union. If he does take a strong line early on, perhaps the most likely time is to insist on free elections in Eastern Europe. It's hard to know just what that would accomplish, though, if the Red Army is still on the ground. I could well see history taking a relatively similar path until about 1946: the Congressional Mid-terms will probably see a different result if Dewey is pressing a reform of the New Deal (and hence provoking a reaction from the Democrats). Of course, Dewey can always confound the Democrats by pressing a moderate civil rights agenda and precipitating a stronger Dixiecrat movement.
Overall, I see a greater potential of Dewey's initial moves to precipitate strong reactions both from the Democrats on the New Deal and from the Soviets abroad. That may in turn create a pretty interesting dynamic to begin with.