Your desire to continue in prayer for others without dependence upon any outward form is itself a good and excellent thing, for intercessory prayer is the sweetest prayer that God ever hears and it is exceedingly prevalent with Him. I deny the saintship of any who does not pray for others, and I would exhort you to abound in this grace, to offer supplications cheerfully and habitually.
Remember that such prayer not only benefits those for whom you plead, but it benefits your own soul. It will lead you to know your brothers and sisters more truly, it will knit your hearts together in love as the heart of one man, and it will fit you to become God's instrument for good to them. It will also bring you often to say, Search me, O God, and try me, and know my ways, see if there is any wicked way in me, and lead me in the way everlasting. Prayer is the most essential thing in turning sinners from the error of their ways, and deep feeling on your part for others shall make others feel for themselves, and God shall bless you, and that right early.
Behold then with compassion those souls that are perishing for lack of Christ, perishing without hope of salvation, and let that compassion move you to earnest prayer and ardent effort for their salvation through Jesus Christ alone, for there is no other name by which we must be saved.
Here is a plain text you may use when you would pray for others, and you may copy it or redact it as you need without any button:
Lord, help me to pray as I ought. O God of peace, I beseech You for those whom You have placed upon my heart. If they have not yet believed on the Lord Jesus Christ, grant them repentance and faith, that they may be saved by Him alone. If they are Your people, cause Your face to shine upon them, revive them again, and make them to live at the highest point of life to Your glory. Forgive their sins, keep them in holiness according to Your will, and conform them to the image of Your Son. Let marriage where You have ordained it be honoured as the lifelong covenant of one man and one woman, and let all lives be ordered in chastity and truth within Your good design. Search me and cleanse me, that I may be a faithful intercessor, and hear my cry for Jesus sake. Amen.
Use this in your closet, in the family, and in the prayer meeting, and agonize with importunity as one who pleads for the lives of others. The Lord who taught us to say, Lord, help me, never finally refuses such pleading, and He will hear and answer for His own glory.