Here is a pretty well known demonstration of syntactic ambiguity in Japanese.
The same kind of ambiguity can also occur in English, as you just demonstrated. So, the "best" way to say them depends on the context. If you have enough context and your goal is to be succinct and not sound too stiff, then perhaps an ambiguous sentence will suffice. However, if you want to be absolutely sure that the sentence on its own will convey exactly what you want it to convey, then it's best to just add the extra context, just like in English.
I think the examples you provided are already as specific as can be using just the syntax, and they all work, except there's still ambiguity:
Last month was my birthday, and today is also my birthday.
Last month it was my birthday (not yours), and today is also my birthday (not yours).
Last month it was my birthday (not my anniversary), and today is also my birthday (not my anniversary).
Today is your birthday, and also my birthday.
Today is my funeral, and also my birthday.
Today is his birthday, and also my birthday.
Today is our anniversary, and also my birthday.
For sentence A, the last two scenarios aren't practical, so there's essentially no ambiguity (however, if we were talking about something other than birthdays they could become viable). Sentence B only has one meaning. But sentence C can mean 3 different things depending on context.