grammar - Why たく isn't used in negative form in this sentence?

言いたくなりますが、見ていませんよ！

I have become a little confused about this sentence since it is not negative and still, in order to say want to instead of たい、たく is used.

Or maybe there was a misspelling when the author was writing the sentence, in which case it is 言いたくない＋ますが

Could it be that this is not, in fact, a negative sentence and たく is used in a positive sense to convey the meaning want to say?

Answer

「言いたくなります」＝「言う」+「たい」+「なる」+「ます」

There is no single "negative" element in this verb phrase.

Among the 4 words, I hope you know that all but the last one 「ます」 need to be conjugated into their 連用形{れんようけい} ("continuative form") in order to form the correct verb phrase. In that process:

「言う」 becomes 「言い」.

「たい」 becomes 「たく」.

「なる」 becomes 「なり」.

And 「ます」 stays the same.

Thus, 「言いたくなります」 is a 100% correct verb phrase meaning "I feel like saying", "(something) makes me want to say", "Every now and then, I want to say", depending on the context.