Yes these can be a little tricky. But great start, both of your examples are right! Whether is more like "if" but with more than one possibly. Always must be accompanied by an independent clause/separate sentence. It implies a condition or something indefinite. Either does not require another clause and does not imply a condition.
Tomorrow it will either be rainy or sunny. --simple sentence
I'm not sure whether it will with be rainy or sunny tomorrow (uncertainty/doubt, condition, like an "if statement)
I believe it will be either rainy or sunny tomorrow (simple statement of two alternatives, possibilities, no condition)
I hope this makes some sense. Try putting the sentence into an IF statement. If it works, "whether" can be used most likely. If not, you should stick to " either ".