Let this be a warning to those intending to change schools, get all your stuff together and
don't let your existing school be tardy. I am sure it is going to cause frustration and
headaches for many others this summer.

Even if you had fielded your offers in February, if your current school is unwilling to provide the paperwork until June, it would make little difference.
The only ways I can think of to prevent this situation is:
1) require that your new employer provides a contract/residency permit that ends the following August.
2) seek positions in the same province; if you change provinces make sure the new employer is prestigious.
3) as previously suggested, subtly refuse to continue to teach and do grades if your current school doesn't provide the documentation in May.

The unfortunate reality is:
1) your current school has no incentive to support FTs leaving; the better a teacher you are, the less they want you to leave.
2) every school has the power to hold you hostage with the release documentation.
3) current regulations make the ability to change schools much more difficult; the government has no interest is promoting turnover and relocation in China: it makes keeping track of FTs more difficult for them.

If your new employer is a major provincial university, they have the ability to convince your current school to be compliant. For the record, in the past 5 years I have successfully changed universities 3 times in 3 provinces. It was an annoying process, but it was done given the influence and proactive effort of my new employers, both were '1st level' universities. I can not speak to the situation at High Schools and private schools.
I hope that your new residency permit does not end on June 30 or you may relive this nightmare. Once you have left, I hope you publish the name of your current employer; they need to be avoided.