The New Testament was built upon the Old. Jesus and His disciples were practicing Jews. He had Passover dinner, debated with religious leaders in the temple, taught in the synagogue... He did not need to teach Jews how to be Jewish because they already were.
Instead, Jesus challenged people --including the leaders -- to get back to the heart of the law after many years of building up too much religious clutter above the basic simple instructions. He broke the Sabbath rules by picking grain to eat -- but He did not really break God's original rules, because it was man that determined that breaking a handful of seeds off a stem was considered work.
Jesus didn't tell us to stop tithing.
Give to Caesar that which is Caesar's; give to God that which is God's. But as KEA44 brought out, the tithe was the first wave of crops produced for the season -- not initially money.
The grain went into the storehouse that was managed by the priests, to dole out portions for the widows, orphans, visitors, underemployed.... it made sure that those who needed to eat, could eat. The priest's families also ate the donations, but they did not hoard it or own it. The grain belonged to the community, and the priests distributed it.
And yes there were donations given to support the work of the priests. But the priests were doing the work of doctors, psychologists, teachers, facilities managers, butchers, lawyers, health code inspectors.... they served the community in many ways.
God owns the cattle on a thousand hills (Psalm 50), so He doesn't need your tithe, personally. Giving is therefore not meant to benefit Him, God uses it as a benefit to us ..
There is a benefit to us to put God first and trust Him; to see everything we have as originally His (and the whole planet); to lower our anxiety of storing up more than we need, and start giving away toward actual current needs; to create a networked community that cares for those in need.
It creates a lifestyle that is generous and good and kind. It honors God when we are not clinging to our money so tightly. It does not honor God when preachers flaunt their belongings in front of donors who gave sacrificially.
The issue is between you and God. Some people have prayed about it and felt they should donate time -- since time is their valued commodity. Or skills, offering pro bono work before starting a project. Or giving away extra belongings to Salvation Army, or volunteering more.
If you take the original tithe instructions literally, then the first visible products of your efforts toward profit -- those belong to the Lord. Maybe that could even be interpreted as offering the first hour of babysitting free to a parent who is struggling. I don't want to mislead anyone, but think it's worth exploring possible parallels to giving grain.
It's God that you are accountable to, so ask Him what would be acceptable to Him. As you say, the tithe went to the poor and you are in a needy situation. The tithe also went toward creating a place for God's presence to be known among nations. Some of the offerings were eaten by the people giving them -- but the act of giving was important.