Aside from the work issue, which I really don't think is a factor since it seems you are most likely working online, I have a few places in which I have spent a couple of months that I would recommend.
* Southwest France - there are many lovely towns in the area you could move in a stay a while. Pezenas, Narbonne, Montpellier - all various sizes and tones. We stayed in Pezenas for five weeks once and I would love to go back. Quite hot in the summer, beaches 20 mins away, and lots of lanes to wander. Many things to do in the area.
* England - I know that's a pretty broad recommendation but parts of the country are idyllic and we've spent long periods there as well. The Lake District - lots of rain but stunningly beautiful with exceedingly picturesque countryside and great outdoor activities. York/Yorkshire - again, beautiful with a mix of larger towns and villages that are peaceful and historic. London - a little pricey but so much to do for you and your family.
* Switzerland - again on the pricey side but we've spent weeks there are a time and it can be done for a reasonable price if you shop for your own food and use public transport. Gorgeous, interesting villages, mountains to climb, spectacular!
* Italy - more affordable. I love the Dolomites, the area around Verona, and Umbria. Again, plenty to do.
For rentals, I would try VRBO or Airbnb.
There is also an interesting website I found recently that may work for you. It's called Trustedhousesitters.com. There are many people all over Europe who are looking for people to stay in their houses and watch their properties and/or pets while they are away, some for extended periods. Click on "advanced search" and choose "long term house sits". I plan to try this after quite a few home exchanges that we have done over the last 7 years.
Good for you for taking advantage of the time you have between houses! You'll be glad you did. Enjoy!