Fifteen years ago when I was looking, I took vacation or sick days off in order to focus on the search. For in-person interviews, I would take “doctor’s appointment” for the morning or afternoon as that office was pretty lenient on that excuse.

Yes, it is tiring and time-consuming; some things have to get sacrificed - I just chose to sacrifice the tasks at my current job.

As for transferring within the same company, if you are a good employee then “the company” is usually happy to have you stay internal even if it disappoints your current team. There is a much lower cost of an internal hire than external, and since you are still an employee your old team can still reach out for support if needed. In other words: talk to HR and see what their policies are. Your new boss will most surely talk to your current boss so it’s not going to be a secret.