When is the closest Conway's Doomsday?
Charcoal, 138 110 98 bytes

≔§⪪”)¶⊟eΦＯ∨ü&amp;-T[¿Ｑ№i⧴⊕%⁰q”⁴ＬΦ⪪”)¶ »Ｒ≦ＰH↘{⎚″4χχβ´ΣP”⁴›ι✂θ⁴χ¹η¿⁼Ｉη¹²⁺⊖…θ⁴1212«…θ⁴¿⁻Ση³η«022§9888Ｉ…θ⁴


Try it online! Link is to verbose version of code. Edit: Saved 28 bytes by switching I/O to compact ISO format (yyyymmdd). Saved 12 bytes when the supported year range was restricted to 1901 to 2099. Explanation:

≔§⪪”)¶⊟eΦＯ∨ü&amp;-T[¿Ｑ№i⧴⊕%⁰q”⁴ＬΦ⪪”)¶ »Ｒ≦ＰH↘{⎚″4χχβ´ΣP”⁴›ι✂θ⁴χ¹η


Take the dates (in mmdd format) 0120, 0317, 0421, 0523, 0623, 0725, 0822, 0922, 1024, 1124. These represent the cutoff points above which the next Doomsday is nearer. (In some cases the Doomsday is equidistant but in particular for 0120 that is not true on leap years.) Work out which cutoff point applies by counting the number of dates that fall before the input date. Then look up the relevant Doomsday date from the list (in mmdd format) 1212, 1107, 1010, 0905, 0808, 0711, 0606, 0509, 0404, 0300, 0012.

If I had access to a date library I could then ask it to fix up my date, but unfortunately I have to do that manually:

¿⁼Ｉη¹²


Is this the 0012 date, meaning the 12th of month 0, i.e. last December?

⁺⊖…θ⁴1212«


If so then output the previous year and a month and day of 12.

…θ⁴


Otherwise the year is at least correct...

¿⁻Ση³


Is this the 0300 date, meaning the 0th of March, i.e. the last day of February?

η«


If not then this is the date we seek.

022


Output February the 2?th.

§9888Ｉ…θ⁴


Output 9 if the year is a multiple of 4, otherwise 8.