Is the phrase "I covenant with You, O Lord, that I will" considered a vow that must be fulfilled, especially if it is said with the intention of encouraging worship?

You must fulfill the acts of obedience you pledged to Allah. If you are unable to, then an expiation for a broken oath (kaffarat yamin (definition)) is incumbent upon you. An expiation is required for each separate pledge, as Ibn Qudamah said: "If one swears oaths concerning different matters... and breaks all of them before performing expiation, then a separate expiation is due for each oath." Pledging to Allah to perform righteous deeds is a vow (nadhr (definition)), which the Prophet, peace and blessings be upon him, forbade. Its consequences can be severe if a person does not fulfill what they pledged to Allah, as Allah the Almighty said: "And among them are those who made a covenant with Allah, [saying], 'If He gives us from His bounty, we will surely spend in charity, and we will surely be among the righteous.' But when He gave them from His bounty, they hoarded it and turned away, while they were refusing. So He put hypocrisy in their hearts until the Day they will meet Him, because they broke what they promised Allah and because they used to lie."