Yes, “J’écris” is the contraction of je and écris. Unlike in English, contracted forms in the French language are mandatory.

J’ is a Je indeed, but it is not the same as the do not/don’t contraction.

In the case of do not/don’t, you contract because you want to save some effort/time by biting one syllable.

In the case of je/j’, you don’t get to choose whether you contract or not. It is a matter of pronunciation: when je is followed with a word beginning with a vowel, instead of pronouncing the e then the vowel, you eliminate the hiatus by removing the e.

In formal language, this is the only case when the e should disappear. In spoken French (or transcriptions, for books for instance), it is sometimes ignored nonetheless, and phrases like je suis are often pronounced chuis and transcripted as j’suis.