Reducing the monthly cost for upcoming months is perfectly possible. There are plenty of alternatives for Azure SQL Database that go for less 400 $/month. Although, the caveat is that they are less able. Since I don't know anything about your workload or how big your database is, I can't say how low you can go. There are editions that are very cheap - but they are only fit for very light workloads.
If you want to take the data out of Azure, about the only way is to extract a bacpac.
As for getting money back for past months, you can open a support case for billing, but I would not have much hope, unless you can show that something has gone wrong on the Microsoft side.