Does the language and structure of "Sonnet 29" by Elizabeth Barrett Browning suggest a healthy love?

That depends. The poem, though very short, equivocates from beginning to end.

It begins with an exclamation that confesses obsession: "I think of thee!" Barrett imagines her lover as a tree. Her thoughts are the vines that wrap themselves around it, not only entwining but enshrouding and strangling it under yards of verdure:

I think of thee -- my thoughts do twine and bud

About thee, as wild vines, about a tree,

Put out broad leaves, and soon there's nought to see,

Except the straggling weed which hides the wood.

Notice that her thoughts not only "twine," but "bud," suggesting growth and potential. They are "wild vines," free and untamed. The object of her love ("the tree") joins her in this dance of fertility by "[putting] out broad leaves," obscuring them both ("soon there's nought to see"), as a couple in an erotic encounter seeks to be obscured. There's only the "straggling weed" which "hides the wood." "Wood," to a contemporary reader, can be very suggestive. Moreover, the weed "[straggles]," it does not strangle, which might be one's first instinct of pronunciation when reading over this poem and contemplating its subject. Barrett, possibly, seeks to trick the eye and, thus, our sensibility: the weed does not constrict the wood, but moves slowly around its circumference.

The "tree" is characterized as a "palm-tree," whose tropical connotations complement the feverishness of love. She appeals directly to her love, hoping it will "be understood / I will not have my thoughts instead of thee / Who art dearer, better!" This can read as unhealthy obsession. Then again, in the context of Victorian poetry and prose, it would not have been. Despite the repression of the era, written expressions of passion -- between both the opposite and the same sex -- were rather common. She ends the last line with an exclamation point, despite its question form. This reflects her certainty in her love.

Her thoughts keep him alive. They are regenerative:

Renew thy presence; as a strong tree should,

Rustle thy boughs and set thy trunk all bare,

And let these bands of greenery which insphere thee

Drop heavily down,—burst, shattered, everywhere!

Here, every line begins with a command. She invokes his strength -- even violence. The tree is strong enough to free itself of the vines (i.e., her thoughts). However, the destruction is palpable. The greenery does not fall, as it actually would, but is "burst, shattered..." Something explosive happens; echoing the sight or sound of a heart breaking, if the metaphor were to come to life.

Yet, the heartbreak is necessary. They both must be free and unencumbered -- detached -- to love more selflessly:

Because, in this deep joy to see and hear thee

And breathe within thy shadow a new air,

I do not think of thee—I am too near thee. 

Once the vine is free, it can "see and hear" the beloved,[a]nd breathe within [his] shadow a new air..." They remain near each other, for the vine is within the tree's shadow. Yet, it is separate, growing not along it, but alongside it. It is no longer necessary to cling obsessively, to "think of thee," for the beloved is "too near."