Mindfulness is a misnomer*. The goal of mindfulness and meditation is to let go of thoughts of the past and the future—to be fully in the present moment, with complete attention on our sensory input. You want to get out of your head and into your body and senses.
“Sensefulness” is an apt term. But what I like even better is “present sense”. It parallels the commonly used term “present tense.” And it reinforces the goal of being in the present moment with attention fully on what you are sensing right now.
Your [thinking] mind is an instrument; a tool. It is there to be used for a specific task, and when the task is completed, you lay it down.
—Eckhart Tolle in “The Power of Now”
* tangentially, mindlessness is likewise a misnomer. People acting mindlessly are often lost in their thoughts and not paying attention to what’s going on around them. They are too much in their mind and barelyin their senses. It's no wonder we struggle to meditate when our language has these concepts completely reversed.