Pray

I often find myself just saying "please help that person with their wordly troubles" when I could be praying for them to get closer with God

Jesus told us that we would always have the poor with us. He also told us to help them. Should we care about hunger, homelessness, addictions, and the rest? Absolutely! But if we only help with physical needs but never address their spiritual needs, we are just making them more comfortable on their joirney to hell. This would still be an epic fail imho.

Yes, we need to pray for their spiritual needs. It could be our fault when they go to hell because we never witnessed to them

Let him know, that he which converteth the sinner from the error of his way shall save a soul from death, and shall hide a multitude of sins. James 5:20
It is quite literally the mission Jesus gave to all his followers. Hospitals exist because followers built them. The Ivy League colleges exist because followers financed them. But as you say, we have the heart to help people survive and thrive and the ultimate of those concepts exist only with eternity in the presence of Jesus. I see the body of Christ, the people not the building, as a place where people should be equipped not just to enhance their own lives, but to equip all to enhance the totality of other's lives. Each of us was created with unique abilities, on purpose. Reaching the needs of others is not a one size fits all thing. If everyone made sandwiches, great! Hunger problem is fixed! People now aren't hungry. Go fix your lives! Doesn't help all. Some ministries provide shower facilities for the homeless, washing their clothes, shelter, cutting/styling hair. Above everything though, the needs of the soul must be paramount.