Is switching to AM5 (7800X3D) crazy right now, or are there better upgrade paths?

Hi everyone!
Before the recent price hikes, I was working on a list of upgrades for my PC (CPU, cooler, RAM, and a new case for aesthetics). With the recent price increases, though, I'm completely stuck.
Here are my current specs:
CPU: Intel Core i5-12400F
MOBO: MSI Mag b660m Mortar Max Wifi
RAM: 32GB DDR4 3200MHz CL16
GPU: RTX 3060 12GB
Storage: 1TB Sabrent NVMe SSD + 2TB Seagate Barracuda HDD
PSU: Enermax 750W
My initial plan was to jump to AMD with a Ryzen 7 7800X3D and move to DDR5 while keeping my RTX 3060. However, given current market prices, buying a full platform upgrade (CPU + AM5 Motherboard + DDR5 RAM) doesn't make much sense for my budget anymore.
That said, my 12400F is starting to struggle, and I still want to upgrade my CPU to give my system some breathing room.

I rarely play AAA games, and when I do, I'm fine with lowering settings. I'm mainly looking to improve performance in FPS/multiplayer games (Valorant, Delta Force, and especially Fortnite, which is very CPU-heavy).
Also, yesterday I noticed a significant performance drop (from my usual 165 FPS down to 90–100 FPS). My CPU usage was unusually high while gaming, while my GPU stayed sitting around 40%.

What do you recommend? Does it make sense to look at a 13th or 14th gen Intel CPU to stay on my current socket (LGA1700)? What would be the better move here? Thanks!
(Also, case recommendations are welcome, right now i have a generic white tower case with rgb fans)