Can not marry or move children after removing mod
Solution 1:
Mods don't exist entirely separately from the base game, such that it's nearly impossible to predict what will happen if you remove them. On PC it might be possible with the developer console to clean up a bit, remove 'stuck' entities (actors) like the children here and force them to move or remove them entirely, but on the PS4 edition it's next to impossible. The way I see it you have 4 options here:
1. Leave the children as-is
It sounds like they are in some sort of half-way state following the removal of the mod. You could simply live with it if their presence is otherwise benign.
2. Re-enable the mod, then try and remove them
Re-enable the mod and load the previous save state, then move them to where you want. Disable the mod again and they should be in the new location.
3. Load an earlier save
Specifically, one before you had children in the first place, or one before you added the mod.
4. Start a new game without the mod
The only real sure-fire way to be free from any of the mod's lingering effects is to have a 'clean' save state without any of the mod's changes.
Solution 2:
Console mods don't have the advantage of a fully-functional script extender and, as such, depend on the game's basic logic to work. This can make mod removal a bit of a bear. Skyrim's default on a failed stat read, to prevent crashing, is to fail silently and bury the exception unless debugging is implemented which is the woe of everyone who has ever made a mod, much less used one.
In the case of the Dragonborn's (DB's) family, there are a few "flags" involved in the base game. These are checked when trying to marry, move, or even in some dialogues.
- Is the DB married? If so, to whom? (Player flag, then NPC flag)
- If proposing, is the target a valid marriage candidate? (NPC flag)
- Does the DB have kids? If so, who? (Player flag, then NPC flag)
- Where is the DB's family home? (Player flag, marked for the NPCs it affects)
- If all of these pass with values, and kids are adopted, associate the familial tags (same things that make Calcelmo mention his nephew died if you kill Aicanthar).
Here's what broke your save.
- Added a mod that increased your marriage options
- Married a character either added or already existed and is now flagged as marriageable
- Got kids, because the player had a home they could go to and a spouse and all of that
- Removed mod, resulting in a corrupted block of data about the family because the person is either not a valid spouse or has ceased existing.
- Skyrim acknowledges the player is still married, but can't find the person, and so it just fails to manage family data properly ever again.
This particular behavior happens because of what I see as a design oversight at Bethesda. Vanilla Skyrim treats marriage pretty much the opposite of how Fable does; you get one spouse, it's for life, no divorce exists, and if your spouse dies the only comfort you'll get is your children's smiles. This is possibly by design to avoid a higher age rating internationally, but also a great way to just hide that the absence of a spouse isn't cleaned up in the code.
So, from Skyrim's perspective, your spouse effectively died, but it can't read your family data properly because of empty references to a spouse you don't have so it just fails or, as likely, punts out a "oh, you don't really have a family" result and skips all following logic.
On PC, there are solutions because you have access to the console commands and can tinker with many of these flags directly. On console, your only real option is to load a game before the mod was installed, or at least before you married your cat. You can try re-enabling the mod first, but this often resets certain flags to 0 (false) for the NPC, especially on relationships, so I don't think that will help. You shouldn't need to restart from scratch; that's more for things that modify gameplay heavily, overhauls, and scripted addons.