Using "though" at the beginning of the following sentence

Solution 1:

Grammatically, though is not the "right option" for the sentence because though is commonly used in a subordinate clause (Though I am hungry, I will not stop for lunch), in a participial phrase (Though bored beyond belief, Jonah continued reading), or as an adverb (Yeah, it was an impressive movie. A bit long, though).

A writer might take liberties and use though in the way that you have (a subordinate clause separated from its main clause by a period) for effect, especially with a long main clause and an impactful subordinate clause (Every morning from then on she would set out from her cabin at dawn to wander through the forest, enjoying the smell of pine and the sweet relief of solitude. Though she never completely forgot Ted.). This is not as common as using coordinating conjunctions in this way (and, but, or...), as you've done at the beginning of your passage. But it is not unheard of.

Alternative words might include: however, still, yet, and other concession words. Still, the preceding sentence tells us that she would "follow us whenever we went", which I presume should be "wherever". For this reason, writing that she "spent most of her time sitting on the wooden steps" seems contradictory, unless the "we" rarely went anywhere.

Solution 2:

The sentence where you used though at the beginning, seems incomplete. All you can do is, combine the last two sentences by removing the full-stop and starting though with a small t.

She would just follow us quietly to whenever we went, like a little stray cat, though she spent most of her time sitting on the wooden steps that led to the beach, gazing vacantly at the blue ocean.

Or you can just remove the though.

She would just follow us quietly to whenever we went, like a little stray cat. She spent most of her time sitting on the wooden steps that led to the beach, gazing vacantly at the blue ocean.

Solution 3:

No, it is not the right option, and while some sentences may begin with though, your "sentence" is not a sentence!

It would be a sentence if it were worded as follows (it includes corrected spelling in brackets):

"Though she spent most of her time [sitting on] the wooden steps that led to the beach, gazing vacantly at the blue ocean, she gradually came out of her shell and began talking to us in very short sentences."

If you want to start a sentence with though, make sure you complete the thought you started with though.

"Though I was not inclined initially to attend the party, I decided at the last minute to go, if only to get a free meal."

"Though reluctant, I acceded to her request."

"Though retired, he kept busy with his hobbies."

This kind of sentence structure works well when you are contrasting two different ideas side-by-side. As you can tell from my examples, after the "though" section of the sentence it is good to insert a personal pronoun. The following sentence, for example, would not be correct:

"Though retired, his hobbies kept him busy."

This sentence gives the impression the man's hobbies retired, and not the man himself!