Time for a proper conversation with your husband, running through a long list of scenarios in advance. Like: will they come over for Sunday lunch every week? No? How many times a week? Call in for tea? Pick the kids up from school and then have a cuppa? Decide what you can both agree on, have a friendly chat about it now, in advance, then talk about what you'll do if in the first few weeks they start calling in more than that/pushing the boundaries you've agreed on.
Come at it from the point of view that you like them, want them to have a good relationship with you all, but are worried about avoiding resentment and the best way is to stop things getting weird.
We have relatives who we have had to negotiate how we engage with them. We have firmly agreed in advance that we will accept no hand-me-downs from their older kids, no childcare, etc etc because we both know things can be turned around on us. We will not rise to the bait if provoked, but we will be ready to make our excuses and leave if needed and will remain vigilant to whether the other looks like they need someone to step in. By having the conversation in advance, it means we both are ready with an 'oh thanks, we've actually already bought one of those' or something similar to whatever is suggested, even if its a suggestion we'd accept from someone else.
You want to avoid a situation where your MIL says something like 'oh while I was in the supermarket I thought I'd pick up a few bits for you, and maybe you can do the same so as to save hassle' and your husband looks at you like, oh maybe that's a good idea? And your choices are cause a scene or go along with it and have some kind of joint shopping list for ever more.
Think through all scenarios: childcare, visits, keys, etc. And start laying the groundwork now. Find ways to get him to drop hints about how you won't be seeing them all the time, if that's what he really thinks. But you need to keep him on side ,and you need to reinforce the idea that you are the family unit, not him, ideally while all is well and everyone is on good terms.
I could probably manage this with my in-laws, but not my parents. I think you need to seriously talk to your husband about what you'll do if you do constantly feel questioned and watched. Would he be prepared to move? And how bizarre is it that they're moving next door but you haven't all had a conversation about why. Its clearly to see you all more, and because they're expecting something - be it more company, more grandchild access, there's absolutely no way they're moving an hour away for their relationship with you to remain unchanged. And your husband knows this. So you need to agree with him now how much it is changing, and where the line is, while you're still in control of the situation.