These phrases are very similar. Often, more than one will be appropriate. I will try to explain the differences.
A. Dry - The most simple. Something wet becomes more or completely dry.
B. Dry out - Something that you is wet throughout(e.g. the tip of a marker pen or clothes that have soaked up water) has become completely or almost completely dry.
C. Dry off - Something has water on the outside of it(often droplets) and the water is being removed, usually to put it in a more appropriate state for what it will be needed for next. Often but not always due to some external action, such as using a wiping with a towel. Commonly used for someone getting out of a pool.
D. Dry up - A pool or resevoir of liquid dries to the point of having no liquid left. E.g. a spill or a river
Based on these we can choose the appropriate option(s) for your sentences.
The spill has become more dry so A is fine.
The spill is on the table not soaked in so B is inappropriate.
Using C implies that the table has done some action to dry itself so this not appropriate.
The spill was a pool of liquid and it has completely dried so D is fine.
I saw that it had dried/dried up
A. The tip is wet and will become more dry so A is appropriate.
B. The tip is wet throughout and will become more dry so B is appropriate.
C. The tip does not have liquid on the outside of it and is not doing any action to dry itself so C is inappropriate.
D. The tip has a resevoir of liquid that will run out of water so D is appropriate.
Put the cap back on the marker or its tip will dry/dry out/ dry up