A Fun Time With Our Dog

So, my family and I went out to the beach, where it was closest to our house.

After my husband let go off her, my husband lead and throw a tennis ball to the sea, and she dashed toward the sea.

Since she also loves running, she started to chaise chase our daughter.

" as if she was a little kid, she tried to escape to our dog.

My husband throw threw another ball to make our dog pay attention to it.

So, my family and I went out to the closest beach to our house.

Actually, our dog loves the water. (just "water" would mean "drinking")

When she looks at some water, she can't stop thinking that she wants to jump into the water and play in it!

After my husband let go of her lead and threw a tennis ball into the sea, she dashed toward the sea.

My daughter and I watched on with a grin.

However, we don't like it because she always shakes the water off her body to drop the water from her.

" as if she was a little kid, she tried to escape from our dog.

My husband threw another ball to get our dog's attention.

However, I really hope we will be able to have more great times together more.

So, my family and I went (out) to the beach that is closest to our house. [This is another way of saying it and "out" is optional.]

We took our dog as well. [I think this follows better, however your English was fine.]

When she sees some water, she can't stop thinking that she wants to jump into the water and play with it! [My suggestion simplifies your sentence a little. I prefer 'see' to 'look' here but technically it does change the meaning slightly.]

So, she was so excited when she saw the sea. [Personally, I think it is hard to determine that a dog is 'glad' to see water. It is often possible to see that they are quite excited so I chose this. The meaning is possibly different to what you wanted.]

After my husband let go of her lead and threw a tennis ball to the sea, she dashed (off) toward it.

My daughter and I looked on it with a grin. [I am more used to the phrase 'looked on'.]

After she caught the ball, she came to us to show the ball and showed it to us proudly. [There are lots of repetitions of the noun and 'to' so I just adjusted the sentence a little. Your English was not incorrect. A more advanced version could be 'After catching the ball, she came and showed it to us proudly'.]

However, we don't like it because she always shakes the water off her body right next to us, so we always end up soaked through. [My sentence includes the next one and I may have used a more 'British English' phrase 'soaked through'.]

We always become wet because of her. [This sentence is fine, however, I included it in the previous one.]

My daughter started to run. [Your English was fine, but my correction already uses 'so'. With this in mind, I removed it from here because I don't think it is necessary in the context of my correction.]

Since our dog also loves running, she started to chase our daughter, [Lots of pronouns may make the sentence a little hard to understand. The pronoun usually refers to the last noun mentioned, which in this case was your daughter. The next sentence be joined to this one.]

who was crying, " Eeek!

" like a little kid as she tried to escape from our dog. ['as if' is fine, however because I use 'as' later on, I changed it to 'like']

My husband threw another ball to draw our dog's pay attention (to it).

It was a fun time for us all of us. [Your English was fine. I prefer this version.]

However, I really hope we will be able to continue having a great time together more.