Out of interest, how do you as an avowed secular humanist define the spirituality that is also important to you?
You didn't ask me, also a secular humanist, but I have an answer. I refer to the spiritual experience, but like you, try to avoid the word spirituality for the reasons you gave - the baggage, including the idea of spirits being involved.
The spiritual experience generally follows some sensory experience like gazing at the night sky or gardening, when we adopt a mindful attitude. We experience connection to our world, with associated feelings such as mystery, awe, and gratitude. There is a thrill associated, a rich sense of belonging and of the world being right and life a privilege. It generally depends on some understanding of what one is experiencing.
The night sky experience is enhanced by an understanding of how far away those stars are and the incredible unlikelihood of a drop of starshine traversing lightyears of space to impact our retinas to inform us of their presence, couple with an understanding that we are literally stardust, the ashes of an earlier star that spread to the nebula that became our solar system through supernova. I don't know if a preverbal hominid could have such an experience.
The same experience can be conjured with a moving passage of music or even a moving church service. I know that when I became a Christian at age 18, my pastor, a gifted and charismatic man, could rouse that feeling in me with beaming smiles, singing hymns, clapping, and the like. In fact, I did mistake that spiritual experience for a literal spirit - the Holy Spirit. Later, upon relocating and trying other churches, I realized that it was the pastor responsible, not a spirit, and eventually left the religion.
I recognize that feeling when I have it today, but interpret it as a psychological phenomenon akin to finding something beautiful or delicious rather than an experience of something external like a god. Just yesterday, a theist on RF was commenting about how atheists don't share the theists experience of God, not looking the right way in the right places. I explained to him that there is no experience of reality available to him unavailable to the average atheist, and that I believe that theists who claim to have evidence and experience of God are merely misinterpreting their experience. This is a typical area where the theist would claim spirituality that he sees the atheist as lacking due to "scientism," which they describe as excessive reliance on science, but which they mean is insufficient reliance on faith. If only we could relax our standards for belief, they think, we could be spiritual like them and experience God, when in reality, if only they would shore up their criteria for belief, they wouldn't mistake their own minds for spirits.
To digress a bit, once there was a time in human evolution when there was no concept of creativity. When people first began having creative impulses, they didn't recognize them as being inventions of their own minds, but rather, something put in their minds by spirits. They were "in-spired." Thus, the muses. If somebody came up with a good poem, dance, or song, it never occurred to him or anybody else that he was its author. Something must have whispered it to him. This is similar, mistaking the output of one's own mind as something experience of external in origin and received.