As with all things ROTC, I yield to those with greater knowledge and experience. This is just one DS's experience.
First, have you received the letter detailing the scholarship offer? Read it very carefully.
In my DS's case, he was offered a scholarship to an in-state school, where he had been accepted. So he accepted that offer - well within the time frame allowed. This was important because if you don't accept a scholarship then there is nothing to transfer. You can accept a scholarship to a school you where you have not yet been accepted, but make sure you have at least applied. DS was fortunate to be accepted at WP as well. You are able to accept both WP and ROTC scholarship while you decide which to attend. He had already decided against WP by the time this story began.
DS made contact with the battalion where he accepted the scholarship. In his case, if the transfer didn't happen, this would be his Battalion, so he didn't want to burn any bridges. He explained that while he was very excited by the opportunity, there was another school that was a better fit for a number of reasons, where he had already been admitted but had not received a scholarship. He told them up front he was going to try to transfer. They wished him luck.
Then he called the battalion where he hoped to transfer the scholarship. DS had interviewed with this battalion and had been in touch with them a few times through the process. He explained what he was hoping to do, and again they wished him luck. Because this all happened with the second board, and before all of the acceptances for either the scholarships already offered, the third board, or the college itself were complete, there were still a number of variables.
He promptly followed the instructions for requesting a transfer included in the paperwork (and called to insure that the request had in fact been received) and settled in to wait.
Rough estimate of timeline...second board results came last week in January. His letter arrived and he put in for the transfer by Feb 10 or so. 30 days were up for those offers around first of March, and third board letters went out fairly soon after as far as we knew. Then those people had 30 days to accept, apply for transfers etc. 5 days after the end of the 30 day period for 3rd board, he was notified his transfer had been approved. His status changed on the portal from winner, noting the original college to winner, noting the new college.
DS did reach out to the desired battalion one more time toward the end of the third board 30 day period. By that point I think the battalion had a clearer picture of how many kids who had accepted scholarships early in the process had actually been accepted to the school, and how many had contacted them to transfer their scholarships somewhere else. At this point, the recruiter was a tad bit encouraging that is was all going to work out, but certainly still non-committal.
As I mentioned earlier, I reached out to ClarksonArmy for guidance. If you have questions I am happy to try to help, but others will have more definitive answers.