What is the biblical basis for both tithing and offering 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 "weightier matters". 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.

I think it misguided to relate tithing to specifically "the old" or "the new" convenants.

google defines tithe as

NOUN one tenth of annual produce or earnings, formerly taken as a tax for the support of the Church and clergy. synonyms: levy · tariff · duty · toll · excise · impost · contribution · assessment · tribute · charge · fee · liability · customs · dues · cess VERB pay or give as a tithe. "he tithes 10 per cent of his income to the Church"

Wikipedia defines tithe as

A tithe (/taɪð/; from Old English: teogoþa "tenth") is a one-tenth part of something, paid as a contribution to a religious organization or compulsory tax to government.

Websters Dictionary defines tithe as

“a tenth part of something paid as a voluntary contribution or as a tax especially for the support of a religious establishment”

I think the first biblical instance of "tithe" (or the giving of 1/10th of ones income) is actually found in Gensis 14:18-20 (long before any Israelite covenant with God at Mt Sinai)

18 Then Melchizedek king of Salem brought out bread and wine. He was priest of God Most High, 19 and he blessed Abram, saying,“Blessed be Abram by God Most High, Creator of heaven and earth. 20 And praise be to God Most High, who delivered your enemies into your hand.” Then Abram gave him a tenth of everything.

In Malachi God says

“Bring the whole tithe into the storehouse, so that there may be food in My house, and test Me now in this,” says the LORD of hosts, “if I will not open for you the windows of heaven and pour out for you a blessing until it overflows.”

– Mal 3:10

The giving of Tithe forms part of a promise by God that is a demonstration of our committment to him. He challenges us saying that in giving tithes to him, we will recieve a pouring out of blessings in return. How can we recieve the benefits of such a promise if we do not uphold our end of it?

Finally, from a logical point of view, I will just provide a personal example (if this forum is willing to accept 1st person statements as evidence)

The religious denomination that i am a member of (and was a former employee of as a high school teacher), use Tithe as a way of providing wages to pastors/ministers and i have no problem with that. In my view its no different to supporting a good cause of ones own choosing. I have willingly chosen to join a religious organisation and I willingly support them financially. Having said that, based on Abram's experience with Melchizedek, and the 14 or so other times tithe is mentioned in the Bible, I do believe that God expects we pay tithe in both the old and new testament times.