Your transmission is taking off in 3rd gear based on the poor acceleration and your engine speed of 3300 rpm at 60 mph at the very end of the video, which is also correct for being in 3rd.
And there is no shifting going on.
Right now, your transmission is in Limp-In Mode (which puts it in 3rd), regardless of the fact there is no check engine light visible on your dashboard in the video, which is a bit odd.
It has either an electrical or a hydraulic problem.
Because you mentioned slipping, the very likely remedy is to pull the transmission and have it rebuilt.
Sometimes you can get lucky with the external electrical connectors but as it was slipping before and is now in Limp-In Mode, the prognosis isn't good.
Good luck.