It's the correct Te Deum, except your "versicles" and "responses" are still part of the Te Deum hymn. The correct versicles and responses are different:
V. Benedicámus Patrem et Fílium cuм Sancto Spíritu.
R. Laudémus et superexaltémus eum in s æ´cula.
V. Benedíctus es Dómine in firmaménto cæli.
R. Et laudábilis, et gloriósus, et superexaltátus in s æ´cula.
V. Dómine exáudi oratiónem meam.
R. Et clamor meus ad te véniat.
V. Dóminus vobíscuм. R. Et cuм spíritu tuo.
Orémus: Deus, cujus misericórdiæ non est númerus, et bonitátis
infinítus est thesáurus: † piíssimæ majestáti tuæ pro collátis do-
nis grátias ágimus, tuam semper cleméntiam exorántes; * ut qui
peténtibus postuláta concédis, eósdem non déserens, ad pr æ´mia
futúra dispónas. Per Christum Dóminum nostrum. R. Amen.
V. Let us bless the Father and the Son with the Holy Spirit.
R. Let us praise and exalt Him above all for ever.
V. Blessed be the Lord in the firmament of heaven.
R. To be praised, and glorified, and exalted above all for ever.
V. Lord hear my prayer.
R. And let my cry come unto Thee.
V. The Lord be with you.
R. And with thy spirit.
Let us pray: O God, whose mercies are numberless and the treasure of whose goodness has no end, we give thanks to Thy most gracious Majesty for the gifts Thou hast bestowed, beseeching Thy mercy, that as Thou grantest the petitions of those who ask, so not forsaking them, Thou wilt prepare them for rewards to come. Through Christ our Lord. Amen.