If you want to make chatting less clunky, at some point invest in a wireless keyboard. Makes a world of difference. Im not a PS4 pro, but some of the cross-hotbar setups seem to work fairly well. Ill leave specifics there for someone who knows better.

As for story vs other classes, thats ultimately up to you. A second(third etc) class will level up much more quickly then a primary class will(due to bonus exp and having more options unlocked), and having more classess leveled up can certainly be an advantage. But as someone with limited available time, getting to new(to you) content sooner would have real value too. So would getting to switch from DPS que to Tank que. Only you know whats more important to you.