How to give my sentimental tulips their best shot at a good run?

I've never been great with plants, but my mom had a greenthumb for indoor and outdoor gardening of all sorts of things! She passed last year, and at her memorial we gave out tulip bulbs and seed packets. So I am very attached to these tulips as a way to honor/memorialize my mother. Thus, while I know nothing is guaranteed, I'm going to do my best with them!

I knew I didn't want to plant the tulips outside, and so I googled several weeks ago and saw that growing tulips indoors requires putting them in a cold place to simulate winter. I saw some people having just bulbs in a cold place, other people having bulbs in soil in cool places. (I don't know if there's a difference between the two other than storage space!)

I put the tulips in a pot of soil and figured I'd pull them out when I was ready to care for them. But I noticed the other day they have started coming up! Obviously they aren't fully dormant in the fridge! lol I'm not sure what I can/should do now.

Due to my sentimental attachment, ideally, if possible, I'd like to be able to propagate the tulips somehow so I can have them with me long-term (I've seen mixed info on if that's possible, reliable, how to do it, etc.-- I've seen things about seeds vs digging up bulbs and replanting them vs. trimming off "bulblets"?) but at the very least I want to be sure they get one good bloom in. They will definitely be indoor-only.

And so what are my next steps to care for these tulips now, and hopefully be able to continue to care for them as long as possible long-term? (I started typing this in the questions thread but thought it was kind of long for it, hope this is okay.)