EN1,
Okay I cheated before by having Dewey essentially take Taft's place and having the presidency all but handed to him in 1908 as what happened with Taft.
What you want is Dewey running as a Democrat candidate and winning sometime after the Spanish-American War, right? IMHO, you need to "get rid of" both McKinley and Roosevelt then. You needn't necessarily kill them off, just get them out of the way politically. Let me explain.
In the OTL, McKinley in 1900 is just too powerful a candidate and almost despite himself. He's the president who just won a war, he's the president who just gained the United States an empire, and he's the president who is presiding over better economic times. That's just too much for even a war hero like Dewey to overcome. In the OTL despite not leaving his front porch again and despite contentions over the Philippines and labor strife, McKinley still beat the frenetically campaigning Bryan by nearly 2 to 1 in the Electoral College. You'll either need to whittle down McKinley's "accomplishments", neuter him politically, or remove him from the race somehow.
Tackling McKinley's accomplishments in 1900 is going to be tricky because they're much the same as Dewey's accomplishments; i.e. winning the "Splendid Little War", and the economy is in the middle of a boom period between the panics in 1893 and 1903. Neutering him politically might be easier, killing off Mark Hanna or having the Republican's divide over the issue of empire might do the trick. Having him decide not to run might be easiest of all.
IIRC, one reason why McKinley twice ran a front porch campaign was due to the frail health of his wife. You could kill her off and have McKinley withdraw from public life. Finally you could kill off McKinley himself. Not an assassination mind you, but just a regular old death. Conroy in his godawful 1901 knocks off McKinley with a heart attack and you could do the same. Perhaps a heart attack after his wife's death or stroke?
If you don't choose 1900, you'll be running Dewey against Roosevelt in 1904 and I think your job will be much harder. Despite there being a mild panic in 1903, Roosevelt's various domestic policies were viewed very favorably and the Democrats had trouble finding a candidate with any amount of national prominence. Even Bryan said no thanks. They eventually ran Parker, but he didn't even carry his home state of New York, which was Roosevelt's too. He only did as well as he did because he carried the 12 ex-Confederate states, plus Kentucky and Maryland, and that region would have voted for Satan before voting for a Republican.
You're familiar enough with TR's career to find times and places to either kill or cripple him. Politically, you could have TR be a little more headstrong, if possible, and the GOP Old Guard more fed up with him earlier to create a split similar to that in 1912 where Wilson won with ~41% of the vote.
Hope all this nonsense helps you somehow.
Bill