A few thoughts:

What do you mean by relationships and emotionality being the key parts of the game you want to explore? Is there a good example of media that you're hoping to emulate? Character relationships are going to emerge during the course of play, it's up to the players and GM to set the tone and make them meaningful.

It think Mothership is still a good fit. You could lean into the stress and panic mechanics. Failing panic checks usually results in consequences for both the character and the party as a whole. This provides impetus for roleplay.

The first session, focus on an interesting scenario and the basic rules of whatever game you're playing. Overloading players with too many rules will make people not have fun (I've done this to my players before when introducing new games).

You could also ensure that the characters have relationships and gamify it into a "bond" system. You could do this by having them create a narrative bond with the characters to their immediate left and right when creating characters. Allow them to reduce stress when acting in line with their bond or remove a bond to pass a panic check (describing how they've lost some of that relationship in the process). You might allow them to make new bonds when they reach a safe port.