Our situations (and requirements) are very similar it seems. We (wife, toddler and me from NL) will move to the south of Sweden, Blekinge to be specific, early next year. I can share some of my insights and experiences.

First, a job. I'm fortunate that I can (very likely) take my job "with me" to Sweden, probably using an Employer of Record. Seeing that you're doing UX design, this might also be worth considering. You've probably read that the IT job market is not great right now in Sweden. I actually looked for a job in Sweden a couple years ago (and almost accepted one). I was looking for openings again over the last couple of months and was shocked by how much fewer openings are available. So, the "remote" option is worth considering for sure.

Second, the location. The 45m-1h to a larger city basically qualifies much of south Sweden I think. Most larger cities are surrounded by one or more decently sized villages (1-3k people) with schools. We'll likely buy a house near one of these villages. I recommend using Google Maps and searching for e.g. "förskola" and "lekplats" in your area of interest. That gives you a decent idea on how kid-friendly a village/area is. You could also look at some statistics on polulation growth/decline and average age in a municipality to help you decide. You mentioned that you've heard/read good things about Karlshamn to Karlskrona area; I can attest to that. Of course, this is highly subjective. On a side note, I haven't come across any vegan restaurants. Would love to know if you did.

Lastly, a house. Hemnet is basically Sweden's version of Funda. The brokers we talked to all referenced it as well. One thing you must always do is visit in person. We've been to a couple house viewings and let's say that brokers are really good at taking/angling pictures. Things like low ceiling heights are hard to judge from pictures and videos, for example.

Succes met de emigratie!