Normally the best time for this switchover is when you enrol for class XI as more number of seats are available at that time.
Anyway in your case you can try to get admission on the basis of your good past academic record as some schools will like to have good students in their institution for improving the overall score of school performance.
Now you find out CBSE schools in and around your town and you may have to meet the principal and explain your situation. You may have to visit may be a few schools affiliated with CBSE. Wherever you get a positive response submit your application along with documents like past educational credentials, earlier school leaving certificate, migration certificate, character certificate, identity proof like Aadhar card etc.
Hopefully, with these efforts, you will be able to get a seat in CBSE school.
Knowledge is power.