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"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) 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", and similar words:

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.

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.

"Pursue" can mean to pursue (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", and similar words:

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.

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.

"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) 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", and similar words:

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.

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.

1
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"Pursue" can mean to pursue (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", and similar words:

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.

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.