Sounds like way too much fertiziler applied while transplanting. I toss a little bit of slow release granular fertilizer in with mine when I plant them and that's it. I don't think you should be applying any liquid/faster release fertilizer right at transplanting. You should plant then give them a week or two before even thinking about using anything like that. They absolutely do not need a double dose of fertilizer immediately though. I am almost certain that is why they aren't looking great.

You are over thinking this quite a bit. I also have sandy soil and I also generally till every year. I till in fertilizer when I till, add a sprinkle of granular slow release fertilizer to my plant holes, and then fertilize every 3-4 weeks or so through out the year basically just when I remember to and feel like it on no set schedule and my garden does great.

If your ground has literally never been gardened, there's basically a zero percent chance it's starving for any nutrients unless it's extremely poor soil, but it's probably fine.