Moving from full frame to APS-C usually feels like a step back in glass quality, and unfortunately, the lens selection for the E-mount crop sensor reflects that. You're right that 56mm is the math equivalent for your old 85mm, but the rendering just isnt the same. I had issues with the Sigma 56mm f/1.4 DC DN Contemporary Sony E wide open; the chromatic aberration was honestly a bit much for professional-grade graduation headshots. It wasnt as good as expected for the hype. If you want something that actually feels like a pro lens:

• Viltrox AF 75mm f/1.2 Pro Sony E is significantly sharper and better built than the Sigma.
• Sigma 56mm f/1.4 DC DN Contemporary Sony E is okay for the price, but the bokeh is kinda busy.
• Sony FE 85mm f/1.8 works fine on crop but the 127mm equivalent might be too tight for an outdoor crowd. The lack of a dedicated high-end G-series portrait prime for this sensor is a major oversight. Its frustrating when the body is this capable but the glass options feel limited...

Saw your post earlier and honestly I got so hyped because that a6700 sensor is absolutely incredible! If you are looking for that 85mm full frame vibe, you have some killer options that wont break the bank. Since you mentioned the budget is 700 bucks, you can actually get professional level glass without sweating it. You should definitely look at these for those graduation headshots:

• Viltrox AF 75mm f/1.2 Pro Sony E
• This lens is a total beast! The f/1.2 aperture gives you such thin depth of field that it makes APS-C look exactly like full frame. Its super sharp even wide open and the bokeh is pure butter.
• Samyang AF 75mm f/1.8 Sony E
• If you want something tiny and light to match the a6700 body, this is it. Its crazy sharp and very affordable, usually leaving you with plenty of cash leftover for extra batteries.
• Tokina atx-m 56mm f/1.4 Sony E
• A fantastic alternative to the Sigma that has a really nice color rendering for skin tones. Its built like a tank too. Personally, I think the Viltrox is the technical winner here tho. The resolving power on a 26MP sensor is just insane. Plus, that extra reach at 112mm equivalent is gonna be perfect for grabbing shots of your niece from the stands if you cant get close to the stage. Seriously, dont feel like you are settling... APS-C glass has come so far lately! I can dig up the MTF data if you want to see the sharpness curves for these.

I actually went through this exact dilemma when I was shooting my sisters wedding last year. I had the a6700 and was terrified the crop factor would mess with my workflow since I was so used to my 85mm on full frame. I ended up trying the Viltrox 75mm f1.2 AF Sony E-Mount and honestly its a beast. It gives you about 112mm equivalent which is a bit tighter than you're used to, but for headshots and outdoor events where you cant always get close to the stage or podium, that extra reach is a godsend. Its built like a tank and the bokeh is super creamy for an APS-C setup. If you're worried about the weight or want to stay closer to that 85mm look, the Tokina atx-m 56mm f1.4 Sony E is another solid shout. I used it for a bit before switching to the Viltrox. It's very light and fits the a6700's compact vibe perfectly. It wont break the bank either, usually coming in way under your 700 dollar limit. Just keep in mind the 75mm has that f1.2 which is just insane for low light or if the graduation runs late into the evening. Its a bit heavier tho, so if you're walking around all day at the ceremony, your wrist might feel it. Both work great with the Sony tracking though, I rarely had it hunting for eyes even in weird lighting.