Some thoughts from down under (we ski in New Zealand and I love demo days) in terms of skis you may wish to look at. I would say the K2s are the right width and 'type' of ski, but anything that is billed as a beginner-intermediate ski will not help you keep up with your family as it will likely get more unstable and chattery at speed, which will put you off even more.
I would personally try and get some demo rentals/demos under your belt prior to purchase if possible - I demo'd skis and also had higher-end rentals for 3 seasons to work out what I really liked in a ski because my previous reference point was generic beginner rental skis and early 2000s twin-tip Rossi Bandits. I'm short (157cm, and anywhere between 50-55kg) and an advanced intermediate trying to be advanced and often ski with my mother (163cm, around 50-53kg) who is a fairly cautious, advanced intermediate skier in her 60s (when I ski with her, we don't avoid any areas on the mountain, but technique certainly takes a back seat when it gets steeper...)
My mother has sold 2 pairs of skis because when she got back into skiing after 15+ years *not* skiing, she focussed too much on having 'light skis that were easy to turn', but sacrificed a lot of stability. She's had the Rossi Temptation 80s 152cm (I think they are now the Experience) back in 2016ish which I thought were noodly and awful, but helped progress her confidence. She then upgraded to 2019 Black Pearl 78, 156cm. With the BP 78s, she liked the easy flex and grip especially on more solid days, but struggled with tip chatter particularly once we had a week of private lessons and she was starting to ski more fluidly and faster. She first learned on straight skis, and I don't think she realised how much she valued having some dampness and stability as like you, she just wanted to get straight on the mountain.
We now have a shared quiver - the 2018 Volkl SC Flair Efficiency 155cm which are narrow carving/groomer skis, but relatively light, and 2019 Volkl Yumis 154cm (titanal band). She skis both fine, but takes the Yumis when it gets steeper as they're easier to turn. I personally like both pairs for different reasons - the Flair tips up on edge easily, but needs a bit of driving and is NOT a spring-conditions ski, the Yumis smear a turn, are very forgiving, and float better in the slush but don't grip as well on crusty mornings.
SO in terms of recommendations, I would think an all-mountain ski with groomer bias or a groomer ski is most likely to suit your needs, and I would look at widths 75-85mm. Here is a list of things I have either tried myself or found and thought the reviews/opinions/descriptions might fit your stated ability and needs. I have assumed that like my mother, you understate your ability somewhat but not your caution.
Rossi Temptation Experience 84 - demo'd this in 2019 and I think it skis similarly to my Yumis.
Volkl Yumi - the 'new Yumi' is supposedly a bit softer and more like the pre-titanal version and comes in both 80 and 84.
Nordica Astral - I actually think either the 78 or 84 would be okay. I demo'd the ??2018 Astral 78 in a length far too short for me on a late spring afternoon so a combination of refrozen slush in the shade, and slush. They were light, easy to ski, but also didn't make me feel like I would die if I went faster which is surprising given that I was on the 144cm length.
30% off the 2019 version here in 165cm.
Head Total Joy (85 under foot,
30% off in the 163cm length here) I believe Head specifically built this series to be lightweight, but have heard on this forum that they can be polarising!
Blizzard Black Pearl - probably the 82. Despite disliking the 78 and the older model of the 88 that I demo'd, these are definitely the Ski Of Choice for a huge number of people in NZ and I see the BP 82 and 88 on the mountain quite frequently. The newer versions are stiffer, which I haven't personally tried yet.
And one out-of-left-field-just-because-they're-cheap suggestion -
ex-demo 2019 Atomic Vantage 86 C - they have a longer turn radius and are relatively long, so probably NOT what you want, but are half the price of every other suggestion I've made!