I've been learning American English. However, I happened to think that I need to learn British English. So I began to listen to BBC. I wonder if there is a problem in learning American English and British English at the same time. One of my worries is that my English might become strange both to native speakers of American and British English.
Don't worry about it.
In the first place, we're a global village now, and in any community you would care to be part of different Englishes aren't regarded as strange or objectionable but merely as an interesting part of who you are.
In the second place, there is a lot of overlap between BrE and AmE dialects. My stepmother spoke pure genteel Tidewater Virginia, and she sounded far more "British" than "American".
In the third place, no matter what English dialect you emulate, what your speech is going to sound like is the English of a non-native speaker, until you are very proficient indeed. —And at this point, revisit the first place. Don't worry about how you sound: as long as your words are intelligible, your accent is part of who you are. Wear it with pride.