"Or are there any other better expressions?" or "Any other better expressions?" sound more natural.
There are a lot of things you could say in this situation. For your first example, the following sounds a bit more natural: "Be careful. Your coffee is about to spill."
For the second example, "drip" is unnatural in this situation. If "drip" were to be used, you might say, "Watch out! Your coffee is dripping!" ("down the side of the cup" is implied)
Other things that sound natural in this situation:
- "Be careful! Your coffee!" (while you gesture to the cup about to spill) This phrase just brings the coffee to your listener's attention.
- "Careful, the coffee's going to spill!"