Telephone registers 'engaged' when trying to dial out. Incoming calls are accepted. Broard band (BT) works. Two atttemps to get help from BT unhelpfull - only comment that the line is OK. So is it the local exchange?
You haven't really established if there is a problem that is your provider (BT) responsibility , via Openreach , to fix, it could easily be your own equipment at fault , and that would be your own responsibility.
You need to try a known working , preferably corded, phone plugged directly (not via a broadband filter) in the master socket test port...using the test port eliminates extension wiring and sockets , and obviously using a different phone eliminates the possibility that your phone is faulty....if you don't have a corded phone , you could borrow another phone from a friend /neighbour/relative , or take your phone and try it in someone else's working phone line.
You can get a corded phone for a few £'s , and keep in handy in case of power failure and situations like this where you need to prove the problem as either your problem , or your providers problem .