What is the biblical basis for both tithing and offering in the New Covenant, instead of just offering? Based on the following passages, it's my understanding that The New Testament clearly supports offering:
2 Corinthians 9 (NIV):

6 Remember this: Whoever sows sparingly will also reap sparingly, and whoever sows generously will also reap generously. 7 Each of you should give what you have decided in your heart to give, not reluctantly or under compulsion, for God loves a cheerful giver. 8 And God is able to bless you abundantly, so that in all things at all times, having all that you need, you will abound in every good work. 9 As it is written:
“They have freely scattered their gifts to the poor;
their righteousness endures forever.”
10 Now he who supplies seed to the sower and bread for food will also supply and increase your store of seed and will enlarge the harvest of your righteousness. 11 You will be enriched in every way so that you can be generous on every occasion, and through us your generosity will result in thanksgiving to God.

Acts 2:44-47 (NIV):

44 All the believers were together and had everything in common. 45 They sold property and possessions to give to anyone who had need. 46 Every day they continued to meet together in the temple courts. They broke bread in their homes and ate together with glad and sincere hearts, 47 praising God and enjoying the favor of all the people. And the Lord added to their number daily those who were being saved.

Acts 4:32-35 (NIV):

32 All the believers were one in heart and mind. No one claimed that any of their possessions was their own, but they shared everything they had. 33 With great power the apostles continued to testify to the resurrection of the Lord Jesus. And God’s grace was so powerfully at work in them all 34 that there were no needy persons among them. For from time to time those who owned land or houses sold them, brought the money from the sales 35 and put it at the apostles’ feet, and it was distributed to anyone who had need.

Acts 24:17 (NIV):

17 “After an absence of several years, I came to Jerusalem to bring my people gifts for the poor and to present offerings.


However, many denominations claim that Christians are expected to give both tithes and offerings, not just offerings (e.g. 1, 2, 3, 4).

Question: What is the biblical basis for claims that Christians are expected to give both tithes and offerings in the New Covenant, instead of just offering?
You asked specifically about the New Covenant, where tithing is mentioned only once.
Matthew 23:23 NKJV
Woe to you, scribes and Pharisees, hypocrites! 
For you pay tithe of mint and anise and cummin,
and have neglected the weightier matters of the law: justice and mercy and faith. 
These you ought to have done, without leaving the others undone.

In other words, they should have tithed and practiced the &quot;weightier matters&quot;. It's up to you to decide whether that means it still applies today or not. There isn't a 100% consensus but most Christians tend to believe that tithing still applies.
As for the difference between tithes and offerings, I'll draw the comparison. Offerings require conscious generosity as you cited above - in contrast, tithing is a prescribed amount. Paying a tithe means literally to pay a tenth. Tithe Etymology
Obviously there's no room for being generous or making such decisions if you're simply paying a predetermined 1/10th amount. The distinction seems clear. The only real argument is the question of Matthew 23:23 itself.