and I don't want to be tied down to working in a single district to be eligible for a 20+ year pension.
What! So you have to work in the same area for a full pension? It doesn't accrue wherever you are?
It is challenging here at the moment. But maybe in a few months it will be better. Here's hoping.
Most teaching jobs are run by the Dept of Education for the respective city and the pensions are paid out by the city. So years accrued in one city do not transfer over to another city unless there is a mechanic for that city to be able to "buy" those years of service based on what you earned there. It is a mess for sure. By isolating the pensions to cities, rather than states or countries, people are tied to those jobs. It is quite unfortunate.
If you have a state job or a federal job, then you would receive a pension from that state or country instead. Such a mess!