There are three straightforward ways to exit Dorthonion:
The Pass of Anach in the southwest, the course of the river Rivil leading to the Fens of Serech in the northwest, and the land of Ladros in the northeast.
(Marked in green below)
Similarly, there are what could be called 'smugglers pass' options (yellow). In the southeast, the mountains that lead to Dor Dínen are not impassable. In fact, we have shown an army of orcs to have crossed them during the Dagor Bragollach. After Ladros was overrun by Glaurung, the orcs passed through Dorthonion and over the mountains, coming around to the Pass of Aglon from behind as a surprise pincher movement.
It's true that Dorthonion also has some forbiddingly tall mountains around it, though they are not impassable so much as...seldom passed
. The mountains around Gondolin are very tall and forbidding, but this does not stop Maeglin from going out at times. I don't recall any part of the story mentioning someone crossing the mountains to the east of Dorthonion; that is (apparently) a fairly desolate place. And, we know that Ered Gorgoroth has steep sides on the southern face, so while it may be possible to climb up into them from the north, climbing down to the south is trickier.
We do have some decisions to make about particular locations in our story. We know that the fortress of Angrod and Aegnor is in the west of Dorthonion, somewhere north of the Pass of Anach. I have been picturing it roughly around the tall mountain above the word 'NU' in Taur-Nu-Fuin. In Season 6, this is now a haunted land of ghosts.
In Episode 2, Beren and Gorlim set out from Tarn Aeluin, and Gorlim goes to Ladros while Beren goes...elsewhere...to try to find people in hiding. Where does he go? He is going to enter a cave and wind up sleeping beside a lake at some point, so any suggestions as to where on the map that ought to be?
In Episode 3, Beren will begin at Tarn Aeluin, and then have to hunt down the orcs who killed the outlaws. In the book, that happens at Rivil's Well (the source of the river Rivil - a mountain spring). Where would we like the death of Gorgol to occur?
Beren will enter the Ered Gorgoroth somewhere under the name 'Dorthonion' on the map. The details aren't super important, but he must be between the rivers, so there is no easy way to get out of Nan Dungortheb...except to go through.