The NT Laws we recognise as being more about 'loving people around us' than the OT Laws, but also, the NT laws are rooted in a culture and a time and a place and issues that the church was facing at the time when those answers were given. They give us the shape of "loving our neighbour" but not necessarily the specifics.
The way the OT law worked was a lot of ancient society wisdom - not eating things that were risky-disky in terms of cleanliness or biosecurity, or which could be mistaken for the religious practices of the peoples around them - AND a lot of stuff that went against the codes of the ancient societies around them. Slaves and servants being granted a rest day off? The sexual abuse of a woman being punishable by either her legal right to your entitlements (as happened for women through the contract of marriage) or your death? Welcoming those who flee their lands of origin?
Those are pretty weird by the standards of the day. And sure, we can argue that 'people should have known better' but we have the experience of some nine-thousand years of human thought and experience and knowledge, transmitted by myth, legend, story, and parable, and then by fairy-tale and epic narrative and collective chant, and most recently by print and recorded media. So, yeah, we know now, but we didn't know then. Just like children start off with very simplistic understandings of the world and grow into complexity with adults. (Sometimes. Sometimes they don't. That's part of the issues we're facing right now: people who want childlike solutions to problems that have been woven by generations of humanity.)
Basically, the OT laws that carry on to the NT, and the NT mandates that we still cling to today, are entirely and without exception those that dignify our relationship with God and with other human beings. We have corrupted them in the modern world with our legalism and our tricksy wordses of online defensive discourse, but in the end, what we are asked to do is to love and care for and succour and support others in ways that lift them up in everything they are, not merely 'spiritual'.
Which is why we don't take the entire literal books of the Torah as our laws. We take the spirit of what they say in them and not the specifics - to respect our fellow human beings as co-created by God, to do right by the created world He gave us to look after, and to do right by the other created beings He put in the world with us, and to give due place and presence to the creator God.
I don't think that we should be focused on "the spiritual" so much as "the spirit".
