I used to often face the problem of speech tags overflowing during writing my papers or creative stories. Their primary purpose (he said, she yelled) is to tell the reader who is speaking. Tags should not be crutches for poorly written dialogue. Sometimes writers use creative tags because they think it adds to the scene. Other writers use tags because, consciously or unconsciously, they know that without the tag the reader will not be able to understand what the characters mean.
If a reader can’t tell from your character’s words and actions that she is angry then the solution is to rewrite the dialogue and description, not to add on, “she said furiously.” Certain words and sentences hold anger. Your job is to create these sentences.
For example, if I wrote, “You know nothing. You never were anything but useless,” you would know that the speaker was upset. Then using physical description, the reader could learn exactly what type of upset. Give the speaker an aggressive stance and clenched fists and we know she is furious. Or the speaker could turn her back and lower her head into her hands and we would know that she is hurt. Either way we are seeing and hearing the character for ourselves instead of having the narrator tell us that she is angry or upset. There are of course exceptions to this rule. The occasional use of another word other than said may be necessary.
For example, if your character gets rid of frustration by standing at the end of a pier yelling his head off, you would need a speech tag that told us he was screaming. Another exception is that dialogue tags other than said can be used to clarify a situation that couldn’t be cleared up by changing the dialogue. If for example, you write a scene where a character who is talking with a friend suddenly sees her ex-husband, then designating that her “Oh my God” is said in a whisper would be appropriate.
If you have too much “he said” and “she said,” then there are other ways to create more variation besides distracting speech tags. One way is to use physical description to make it clear who is speaking. Another way is to use a so-called "live dialogue" where it is clear to whom certain remarks belong even without tags.
Useful Resources:
Know The Worth of Your Writing!
Simple Revising Techniques
What Makes a Good Children's Story?
The Four Firsts-Point of View
Building a Strong Story Structure