Does anyone have a recommendation for a good bike lock and were you ever in a situation where you didn't have an ideal spot to lock your bike?
Much has been written about bike locks with very little consensus. I think everyone agrees that it depends on where you are. In some places and circumstances, no lock or a small lock is sufficient. In others, the best lock in the world is insufficient. Everybody gets to pick their own balance point between risk, convenience and weight. Some people take no lock at all, some a very lightweight cable, and others heavy chains and U-locks. Take your pick.
How did you deal with washing your clothes each day/dealing with them being damp for awhile?
I think most people look for a Laundromat periodically. Others just rinse them out in a sink or shower. In many climates, they will dry overnight on a line. If not dry, you can strap them to the outside of your gear. Don't pack damp clothes inside of waterproof panniers.
if anyone has a recommendation for a very light weight easily packable tent
There are many suitable tents on the market. A lot of people subscribe to the N+1 theory, i.e., buy a tent designed for one more person than you have. Minimalists go with tarps or bivvies or hammocks. You can get tents under 3 pounds, but they tend to be quite expensive. Tents over 6 pounds for one person are too heavy. You might consider something in the $200 range at about 4-5 pounds. That's a nice compromise. Many tents fit these parameters. If you are a true minimalist, however, spend twice that much to get the smallest, lightest tent (or just take a bivvy), and skip the ground sheet.
I also am thinking of how I will go about packing for this trip. I don't want to use a trailer. The only things I figure I'll be carrying are toiletries, a tent, and water/snacks plus spare tires and general necessities. Should I use panniers or what do you guys think the best way for me to go about packing my bike is with the goal of being as light as possible?
Most people don't go for "as light as possible". But some do. Again, there are tradeoffs involved and you get to pick. It depends on what you mean by "as light as possible". It's possible to go quite extreme, especially if the expected weather is mild. Cutting weight of gear means that you can cut weight of the panniers and racks too, so the effect multiplies. High-quality waterproof panniers can be quite heavy, so if you're cutting weight to the bone, you probably want simple, lightweight panniers with plastic bags inside to keep things dry. Never take two of something if one will do. You can get by on just one set of riding clothes and just one set of off-bike clothes. Figure out what the coldest conditions you will likely experience and then take just enough clothes that if you put all of them on at the same time, you won't freeze to death. If you're going super ultralight, you probably won't want to take a spare tire. You may be able to get by without a spare tube either, but that will leave you vulnerable to the small risk of an unpatchable failure. A tire lever and a couple of allen wrenches are the only essential tools. Again, every decision comes with some level of risk.