Planning for new builds mortgage tax increase

Hi all! I live in CO and moved into a new build last year. We received a tax assessment in June estimating our upcoming tax increase to be about $600 more a month. We also have a possible law coming up in CO on the ballot to decrease the percentage of the tax reassessment because at the same time we were being reassessed with the expiration of a law that protected tax hikes as well.

All that being said, I don’t actually know if it’s going to hike up that much for sure. I’ve seen in many posts in this sub of people that have been shock to find that their mortgage went up by $1000/month due to “escrow shortage” when being in a new build for awhile. My question is, is it a good idea to just start putting an extra couple hundred into escrow account now every time I pay my mortgage to be ahead on that? Does that come back to me at the end of the year if I overpaid? Are the people that are posting this getting charged that much more because of fees or is that just to catch up for the previous year probably? The tax hike will be in effect in 2024 I believe so I have some time to plan for it. And possibly get ahead. Thanks for any advice!