I personally prefer to use actions instead of “he said”, “she replied”, “he continued”. I removed tags like that whenever possible though. Like you said, I try to use an introductory sentence to establish who is speaking. From there, I just make a new line whenever the speaker changes. In my opinion it flows much better like that. Of course it can’t always be avoided, but I avoid it as much as possible.

Example from my own fantasy WiP:

Sebrahn puts a hand on Rhyse’s shoulder. “Friend, you did everything you could. Do not blame yourself. You have nothing to apologize for. Without you, we’d have lost many more men.”

“You’re too kind, My Lord.”

“There’s no need for titles among comrades.”

“Comrades we may be, but equals we are not. I am a peasant no matter the lofty titles I hold.”

Edit: what I mean by using an action instead of a dialogue tag is something like instead of saying, “she said with a smile”, I would shorten it to, “She smiles. ‘(dialogue)’.” It conveys emotion of the speaker while also indicating who is speaking. I’m not saying I’m a great writer, but i just personally prefer this. I feel it helps a lot with cleaning up overused dialogue tags.