With a numerical superiority and a technical inferiority, would the Russians have done better to opt for a battle of attrition instead... perhaps using a line abreast formation?
They tried. At 12:15 in response to sighting real live Japanese torpedo boats, Rozhestvensky ordered his First Division battleships into line abreast. Imperator Aleksandr III turned the wrong way throwing the formation into chaos. Rozhestvensky canceled the order and returns to line ahead.
This has consequences for the entire battle. When firing begins, the Russian fleet is still trying to sort itself out from his botched maneuver. The lead battleships mask the trailing fleet. Oslyabya, leading the second squadron, has to come to an almost full stop to avoid colliding with the trailing ship of the first squadron, Oryol, and takes an almighty battering.
The 2nd Pacific Squadron could have done more to prepare and to avoid the battle entirely. The squadron was hastily assembled out of whatever ships were available at the time to break the siege of Port Arthur, but the port had fallen well before Tsushima. Their new objective was to reach Vladivostok and link up with the remaining ships there. After rest and repairs, the combined fleet could recommence operations against the Japanese. Now that the pressure of relieving Port Arthur was gone, haste was no longer necessary.
Supply more ammunition for practice
The 2nd Pacific Squadron was hastily assembled out of whatever ships were available at the time to break the siege of Port Arthur. Rozhestvensky was an experienced and effective leader, but his crews and subordinate commanders were not. They could have used the travel time to practice, but the squadron was supplied with limited ammunition which opportunities for training. Given this was the last throw of the dice for the Russians, this was a strange time to be stingy.
Leave the obsolete ships at home
Rozhestvensky was saddled with the slow and obsolete Third Division consisting of three coastal-defense ships and the even slower Imperator Nikolai I. These ships had little impact on the battle and only served to slow the fleet down. Their slow speed meant the Japanese could fight at their desired range. Without these ships, Rozhestvensky would have had more options. With these ships, once spotted he would be hard pressed to disengage from the Japanese.
Take a different route
With a need for haste gone, the 2nd Pacific Squadron could take its time reaching Vladivostok and avoiding battle in their dilapidated and divided state. They could have avoided the restricted waters of the Korea Strait and taken a longer route around the eastern side of the Japan. Rozhestvensky telegraphed his intentions when he dismissed his colliers; without them his ships had no choice but to go through the Korea Strait.
Use the Oryol as a ruse
The rules of war required the hospital ship Oryol to remain well lit at night. She was spotted the night before the battle by the Japanese auxiliary cruiser Shinano Maru who investigated and spotted more Russian ships nearby. Thus alerted, the Japanese began shadowing the Russian fleet.
Knowing the Oryol would have to remain lit at night, Rozhestvensky could have sent her off on her with some fast auxiliaries off on their own as a ruse.
React to the scouts
Long after it was clear they had been spotted, Rozhestvensky continued to maintain radio silence. Japanese scouts were allowed to shadow the Russians and radio reports back to their fleet with impunity. Visibility was poor, and had Rozhestvensky been aggressive against these scouts he may have been able to slip away.