Overcontributed my 403b/401k two years in a row. What now?

I didn't realize the cap applied to both 403b and 401k accounts. I maxed out my 401k and then put about 5k in my 403b last year and about 15k in it this year. I asked my employer (a huge educational institution) to take out my overage this year before April 1, but I don't think they are going to do it in time. They keep telling me to wait, and it's getting close to the deadline.

So what happens if they don't? Should I withdraw all the overages in 2025 and take the penalty? Should I keep them in? Will the IRS really figure out my overage in 20 years and assess me a huge penalty when I retire and try to withdraw? Or can I just gamble that nobody will ever notice?