Can I do spellwork without burning anything?

You can treat the fire as one possible symbol, not a required ingredient. Fold the paper and tuck it somewhere private, tear it into small pieces, or recycle it when you feel ready. The meaning can come from the intention you set, rather than how the paper is disposed of.

If the point is letting go, write the intention down, read it once, then shred or tear the page and put it in the recycling. If the point is keeping the intention present, put it in an envelope or a journal instead. Pick the option that feels comfortable, not one that makes the ritual stressful.

There’s no need to use fire if you’re uncomfortable with it. You could fold the paper and leave it in a box, or soak it in a little water and discard the pulp in the trash. I’d avoid putting paper or other materials outdoors, especially if they could become litter. And a story about a ritual leading to a relationship doesn’t establish that the ritual caused it, so keep expectations grounded.