The scar scene, where Astarion tries to figur out what they mean, triggered for me relatively late, when we were already in the underdark. I went to camp much more often than it would normally make sense (fortunately, a long rest is not necessarily needed for most of the dialogue scenes).
I had not used the tadpole and I had sided with the Tieflings, spend the night with Astarion and let him bite my character, and after the party, the dialogues had triggered for Astarion first.
I had Shadowheart in my party as well, but I didn't have the problem that her scenes overwrote his, although Shadowheart's approval was slightly higher than Astarion's (both had high approval, although I played a kind and helpful character, and I did
not let Astarion kill Gandrel, of course).
The dialogue scenes I had with Astarion so far were:
- first night at camp
- discussing our preferred methods to be killed (this scene sometimes triggered only for Shadowheart, but in some games, I had this dialogue with Astarion. It seems to trigger for the companion with the highest approval)
- the stargazing scene (it seems to be triggered after you have talked to Zorru about the Githyanki patrol, but before you talk to Nettie. Sometimes, this scene triggered for Shadowheart, when her approval was higher at this point)
- finding the dead boar, which triggers the bite scene the next long rest. (I let Astarion bite my character, defended him when the other companions were sceptical to have him in our party, and told him he could bite our opponents in a battle)
- spending the night with Astarion after the party (first time we notice the scars)
- wondering how the blood of the other companions taste (a mere thought experiment, of course)
- Astarion had a little too much bear blood
- the mirror scene (it would be nice if we could commission a portrait of Astarion when we meet Oscar Fevras again)
- the scene where Astarion tries to figur out the meaning of the scars and reluctantly accepts some help. My character was not a Tiefling, so she couldn't read infernal, but at least she could draw the glyphs.
I am sorry I cannot be of more help. The one thing I noticed is that after the party, and after starting the romance, it seems to be less important who has the highest approval.
I had the same issues with Gale's romance, some scenes did not trigger for me, either, but it was much better since the last patch.
But I hope the companion dialogues will be more easily accessible on full release, too.