Both i-adjectives and na-adjectives conjugate in Japanese. The conjugation form that allows adjectives to modify verbs (and other adjectives) is called 「連用形{れんようけい}」 ("continuative form").
With na-adjectives, the 連用形 comes in the form 「〇〇に」, which means that you only need to change the final 「な」 to a 「に」.
しずかな人{ひと} = "a quiet person"
しずかに食{た}べる = "to eat quietly"
Fairly simple, isn't it? You can expect to use this rule all your life, too.
元気{げんき}なこども = "a vigorous child"
元気に生{い}きる = "to live vigorously"
Please remember that there is no such thing as the te-form for na-adjectives. Moreover, even with i-adjectives, it is misleading to say that the te-form modifies verbs. It is again the 連用形 that modifies verbs for i-adjectives as well.
For the 連用形 of an i-adjective, simply change the final 「い」 to a 「く」.
うまい歌手{かしゅ} = "a good singer"
うまく歌{うた}う = "to sing well"
It is utterly incorrect to say 「うまくて歌う」 and I certainly hope you did not learn that it is correct.
高{たか}い空{そら} = "high skies"
高く飛{と}ぶ = "to fly high"
We never say 「高くて飛ぶ」.