You go home and just start applying. Dress up your resume first.
Keep track of each job you apply for in a spreadsheet and forget about it. I find this helps keep me from getting discouraged. Half the time, when I get a rejection email, I don’t even remember applying for the job. My philosophy is “it’s a ‘no’ until it isn’t”.
You say nothing about it until you turn in your resignation. Especially to the “gossips.” That’s just feeding the trolls. If you’re looking for an internal transfer, I wouldn’t see why that would be a problem. That happens all the time. If you work someplace where that is a problem then that means you work for a toxic shit hole and should seriously consider leaving.