"Get" is not necessarily incorrect, but it should be considered more informal compared to pursue, and you should probably use **get with**:

> He is trying to get with me, but I am not interested in him.

"get me" wouldn't be incorrect grammatically, but it's a little awkward and ambiguous about the romantic sense. This is because while people say "Go get her!" to mean to pursue another person, it's rarely used to refer to yourself (in my experience). Instead, "Get me" is often used in the sense of "get me **to**":

> He is trying to get me to do something.

"**Pursue**" can mean to [pursue (1)][1] someone romantically or in terms of romantic pursuits:

> He is trying to pursue me, but I am not interested in him. / He is trying to pursue me *romantically*, but I am not interested in him.

**Woo** means "to [sue for the affection of and usually marriage with](https://www.merriam-webster.com/dictionary/woo)", and [similar words](https://www.thesaurus.com/browse/woo?s=t):

> He is trying to woo me, but I am not interested in him.

**Woo** is more formal than **pursue.**

If you want to go more simple and straightforward, you can say "**wants to date**":

> He wants to **date** me, but I am not interested in him.

This is also generally interpreted to be in the romantic sense:

> He wants to **be with me**, but I am not interested in him.

These are more common in spoken English.

or even say he's interested and you're not:

> He is interested in me romantically, but I am not interested in him.

  [1]: https://www.merriam-webster.com/dictionary/pursue

Edit: I saw your edit to your question. You could add **would do anything to** to indicate desperation as in:

> * He would **do anything to be with** me, but I am not interested in him.

> * He would **do anything to date** me, but I am not interested in him.