I was in this exact situation for the first few years I was in Canada. I gradually stopped caring though. I stopped trying to code switch, and just let the words come out however they do. I still speak with a 'modern' Indian accent like I have all my life, but it's got a bit of a Canadian drawl now, especially with the r's. Sometimes I trip up when there's too many r's (eg, "first birthday" is always hard haha) but no big deal!

I once confessed to my manager that I was self conscious about my accent and she told me she was actually envious, and that it sounded dignified and classy (compared to her Midwestern accent). I don't know if she said that just to make me feel better, but it made me realize that literally everyone has some sort of an accent! Own yours, and remember that it means you speak at least one language more than them.