we need to change over to the Julian calendar

That's where the names of the months of the Gregorian calendar (which is what we're using) come from, the Julian calender got them from the old Roman calendar (which was inaccurate as fuck). The main relevant change in the Gregorian reform is the spacing of leap years, making it drift less than the Julian one. It's still drifting a bit and a fix was proposed back in the 19th century but never adopted. We'll probably revisit the topic in the decade before the year 4000 where it's actually going to matter.

If you want a sane calendar try the Discordian one. Though arguably St. Tib's day shouldn't be right in the middle of a season. I'd suggest considering it the 0th day of Chaos, giving an additional hangover day for New Year's every once in a while, also, set the St. Tib's day years to the same stuff as the aforementioned reformed Gregorian, whith an asterisks saying "change as needed once the earth starts falling into the sun for real". The starting year (1 YOLD is 1166 BC) is fine because it's completely random.