cross : We use across as a preposition (prep) and an adverb (adv). Across means on the other side of something, or from one side to the other of something which has sides or limits such as a city, road or river:
Example :
1. We took a boat across the river. [PREP]
2. My neighbor came across to see me this morning. [ADV]

Over : We use over as a preposition and an adverb to refer to something at a higher position than something else, sometimes involving movement from one side to another:
Example :
1. From the castle tower, you can see. [PREP]
2. Suddenly a plane flew over and dropped hundreds of leaflets. [ADV]