How do you actually break into a pileup when 15m sounds like this?

What you are running into is just the nature of a contest band during peak activity. It is a madhouse, basicly. The primary thing to understand is that there is no single signal coordinator telling everyone it is their turn, so you have to become a skilled listener. To answer your question about knowing when it is your turn, you don't just wait for a 'QRZ' and nothing else. You are looking for a distinct pause, usally after the DX has finished their transmission and before the noise floor or the next set of chatter begins. If you are calling into clear, and no one responds within five or six seconds, then you are likely clear to transmit.

If you are getting shouted at, it usually means you are breaking in while two or more stations are mid-conversation, which is the most common mistake. Don't assume the station you are talking to is the only one; there might be a third or fourth station listening and waiting to chime in. When you hear two stations calling each other back and forth, the moment they both pause is your chance. It takes a lot of patience to learn this cadence.

As for stations moving up or down the band, that's totally normal on SSB. Stations often jump frequencies to find more open air or to check different modes, and you absolutely should follow them. If a station is moving from 21.2 to 21.4, you move with them. If they move up to 15m, keep following the thread until the signal cuts out or they make a clear change of plan.

Since you are just starting out and want to log something, focus on calling CQ when the band is quiet, which is still challenging during a contest but often yields a response. Also, don't underestimate the quiet moments; sometimes a lone operator who isn't part of the main chaos will pick up your call when everyone else is too busy talking DX.

It sounds like the whole thing is really dependent on timing and just listening carefully. I feel like I'm going to be really awkward when I finally get a chance to call someone. Do you have a specific way you say hello when you start talking to a station? shold I always say CQ even if I see someone operating and I just want to make contact with them? I'm worried about being overly formal or making a mistake.

I remember when I first got serious about this, I was just staring at the bands feeling lost. The whole process really clicked when I started using an old audio analyzer app on my phone. It let me see the frequency shifts and how fast someone is moving around. It showed me wich spots were getting crowded versus which had a bit of breathing room. That really helped me get comfortable with the movement across the band. It felt less like a guessing game and more like tracking a moving signal.