Unlikely, since while the Russians were not exactly a strong naval power; knock-out battles like Trafalgar were rare. Well, let's go with that type of blow-out anyway: It'd completely stuff the Russians options since Japan could've been free to resupply and move material without much worry (as opposed to it being done in bursts) until the Baltic Fleet came to try and dig them out. This would likely have given them the ability to KO Port Macarthur quickly and basically win the war on the ground, since their supply lines were much better than the Russians should they be able to achieve full naval dominance.
So with that in mind, I'd say a Japanese victory in the 1st Russo-Japanese War, rather than the painful draw which led to the spheres of influence the two nations would later on redress in the Great War and the far more decisive 2nd Russo-Japanese War.
//This is my first DBWI. It's...weird.//
That would definitely mark Japan as a Great Power, and not just another, non-white nation in Britain's pocket. And speaking of Britain the British were very allergic to anything that shifted the balance of power. A new Great Power on the rise, Russia effectively destroyed as a naval power (seeing as the Black Sea Fleet wasn't really much of a threat to anyone apart from the Turks), definitely counts a 'shift' in the BoP. How would they react? And how much could the Japanese extort from the Russians? Sakhalin and Kuril almost certainly, Korea probably, but what else? And how about the rest of the world?
The thing is, they really, really, really hated Russia at that time due to the "Great Game" they kept playing in Asia. I imagine that they wouldn't mind there being a new balance of power in Asia, as their main concern was Europe due to the fact that if a powerful continental power came to being, that might be able to strangle the home islands. They however would probably quickly grow to try and counter Japanese Imperialism if it began to interfere with things like the agreements and influences they had in China for example.
...and the far more decisive 2nd Russo-Japanese War.
Very decisive indeed. The Second Russian Republic still claims the Maritime Province, even if they've given up on Sakhalin and they've recognised Manchuria as part of China. Of course, the Maritime Province is so thick with Japamese settlers that it'll never go back to Russia...
Regarding what might have happened... In some ways, the drawn war worked out OK for Japan long-term. The Russians recognised Korea as being in Japan's Sphere of Influence, but saved face on that score by insisting on Korean independence being guaranteed. OK, the treaties Japan had with Korea up till 1948 gave them massive influence over the peninsula, but the treaty meant they never tried to be too outrageous (or push for annexation, like some Hawks wanted). That, coupled with Japan and Korea fighting on the same side against Russia, led to the situation we have now: Japan and Korea as allies, 'friendly rivals' and regional leaders.
The Fall of Seoul for instance: over 200,000 Japanese and Korean soldiers dead, including General Kuribayashi (though he took 500,000 Red Army soldiers with him), followed by a week of slaughter, rape, looting and other depravities on Seoul's populace by the Red Army.