Clean off any makeup or hairspray residue with Krud Kutter. Scrape off all that loose paint. Sand, sand, sand. Prime with BIN shellac based primer. Another primer coat. Light sanding with a high grit. Apply BM Advance, 2 coats, drying 24 hours between, and let cure 30 days before being subject to hard use.

I would look into getting a new bathroom vanity. Find the cheapest one on sale that will fit. If something's looks nice it won't be cross examined.

For the kitchen, I guess you didn’t get touch up paint kit? But if know the color you can just buy something to match to blend in at the paint store.

I agree about replacing the bathroom vanity. Repainting is possible, but I imagine you would have to repaint the whole thing to get it to look right. I also imagine that you have enough to do to prepare for a move, and you don't have a spare 20 hours or so to deal with the vanity. Replacing is a DIY job that can be done in an afternoon (including going to the big box store to pick one out).

I'm not sure I'd do anything with the kitchen cabinet - looks like normal wear and tear to me. How noticeable is the scratch from 4 feet away?

How much water are your kids getting on the cabinets? I would be looking at what they do and how you can change/improve the behavior/habits so they aren't causing so much damage.

I have never had bathroom cabinets look that bad, even ones used by kids for several years.

I have no idea. How that happened. They usually dry the counter right away but I am so regretful what I have bought. Any other suggestions should I call the shop where I got from?

10 years old will be a useless call to the shop. I would replace it. Stripping it well enough to repaint would be more work than replacing it.

Removal, replacement and plumbing would be a lot of extra cost, and getting a cabinet to fit the space perfectly will be a challenge and could have ramification with the tile floor, countertop, trim and walls. Try not to open a can of worms. I would repaint the cabinet according to HU-0228123141598721's instructions. Remove the cabinet doors and drawers to work on them and give them time to dry.

For the kitchen cabinets, buy a touch up crayon in a white that closely matches your cabinets. Even if it isn't an exact match, it will still look better than the dark scratch.

Since there is only a wall on one side of the vanity, and assuming the current one is a standard size, finding a replacement would probably be less time and effort than repainting what's there. The entire vanity would have to be painted, not just the door and drawer fronts (unless the OP went with a contrasting color).

Here's a possibility:

Bathroom vanity

Please suggest what kind of crayon wood paint? I have no idea that we crayon wood paint now,

Saleha, this is the brand that I use. It came with my cabinetry. It's not crayon wood paint, it is a touch up (crayon) stick. You just fill the scratch with the crayon and then wipe off the excess. If you make a mistake, no problem, just do it again until you get it right. Easy peasy.

Thank you so much!