The habit that ends this problem permanently: stockpile in autumn.
Leaves are free, they arrive in enormous quantity, and they are the ideal counterweight to a winter of kitchen scraps. Bag them up in autumn and stand the bags next to the bin. Two or three bags will carry a household through to spring, and it takes one afternoon.
If you missed this year, cardboard is genuinely the best substitute and you can get as much as you want from any shop that will let you take their flattened boxes.
Two things that make everything work faster regardless of material:
Surface area beats the list. Shredded or torn material breaks down several times faster than whole. A whole box takes months; the same box torn into palm-sized pieces takes weeks. If you are willing to run leaves over with a mower before bagging, they compost far quicker too.
Mix, do not layer. The layered lasagne picture in guides is for building a pile in one session. For a bin filled gradually, every addition of scraps should go in with a handful of carbon on the same day, mixed in. That single habit prevents the sour wet state entirely, and it is much less work than fixing it afterwards.