I have some experience with a chainsaw, and I am looking to purchase an Alaskan sawmill attachment and rip chains for my Stihl MS311.
You can chain saw mill, but this is much harder than it sounds and is very slow! Get a local Sawyer if you can, as few that try "chain saw milling" without an extensive background in "green woodworking," and other intensive wood based skill sets can achieve good or sustainable results.
I am fairly certain the milling/ripping will get very difficult if the wood dries, but how much time do I have?
Depends on the species, but full logs do not dry out that fast in general and much slower when "end sealed," which you should do any way as that is proper wood management. Get the bark off as soon as possible as this is a major contributor to insect infestation. If it does start to warm up on you, you may have some sap stain issues. The logs should also be at least 150mm off the ground (300mm is better.) Store in shade if possible, under water is best, and again...end seal.
What is the best way to go about storing this lumber until I need it, if I can't have the ideal drop-mill-build scenario?
Do a web search for proper "stickering" of lumber, then if you have more questions come back.
But it is important that you get them down before the sap rises which I am guessing is probably starting there. Winter harvested wood from dormant trees is drier for initial moisture and you will have less trouble with cracking later.
Sorry this is an "old wives tale." There is the same amount of sap, summer and winter. The sap does not fall. Many forests are cut winter and summer. There are different precautions for each and weather can have affects as much as anything else, depending on goal for the finished wood. Winter is often easier on loggers, and the ground they tread upon. "Sap Rot" or "Sap Staining" is mitigated in winter months.
Good luck,
j