I am obnoxious to each carping tongue Who says my hand a needle better fits.
Here's another little switch in tone. The speaker goes from worrying about her own shortcomings to going after her critics.
Clearly she feels like she's under attack.
When she describes herself as "obnoxious," she means "vulnerable" or "exposed" (not obnoxious like your little brother).
These complaining voices (each "carping tongue") say she should stick to her sewing.
This is basically the seventeenth-century version of "get back in the kitchen" and Bradstreet's speaker is having none of it.
Rather than ignoring prejudice or giving up her poetic career, she's going to use these lines to directly confront any potential male critics. That's a pretty powerful stance. Whether or not she thinks she's as good as a man, she's not going to give up writing to make these people happy.
A Poet's Pen all scorn I should thus wrong, For such despite they cast on female wits.
The syntax is a little complicated here, but basically the speaker's saying that her male critics (who might be real or imaginary) have nothing but scorn for the idea of her writing poems. They worry that she will "wrong" (mess up, violate) the craft of poetry. (Line 26 is basically saying "Everyone thinks I'm going to royally screw the art form up by trying to attempt it.") Their reasons for thinking this don't have anything to do with this particular female poet. They show contempt ("despite") for women poets in general, and especially for "female wits" (that could mean either "female intelligence" or "female writers"). Jerkstores.
We should point out that not all men felt this way about female poets. In fact, a bunch of Bradstreet's male relatives helped her to get published. On the other hand, they didn't ask first, and they made it clear to her readers that she'd finished all her chores first. Sigh. It wasn't easy to be a female poet in seventeenth-century New England.