I have two bicycles. One is a 3-speed Raleigh cruiser from the 70s that my dear husband got me via Craigslist. I love this bike. It is very cushy, comfortable, and stylish, and is great for riding around town. However, I do not want to use this bike for my commute as it is 1) heavy, 2) only has three gears, and 3) is old and made of steel (I think) and I really do not want to get it wet. It's just for riding around for fun.
It has been mentioned that both of your bikes are made of steel, so the following applies to your Trek as well. Rust is not a problem for steel frames, as long as you take some small precautions. You are correct in thinking that wetness leads to corrosion, but rust takes a lot longer to form than most people realize. If your bike gets wet on the way to work, sits out all day, and you dry it off when you get home that is absolutely no problem. Any corrosion that may have taken place during the day will be so small you won't even be able to see it. The big problem is when water gets inside the frame unnoticed and sits there for months. That's when your frame will rust out.
Keeping the outside dry is super simple. Anything that is painted will not rust, and doesn't really
need drying, though you probably still should on principle. Anything that is not painted (chain, sprockets, handlebars, any metal exposed by scratches in the frame's finish, etc.) should be dried within 12 hours of getting wet (just towel it off). All the components in your drivetrain should be oiled regularly (once every hundred miles or so), and this oil will help hold off the rust. You can oil or, even better, wax the remaining nonmoving steel parts for a little extra rust protection.
Keeping the inside dry is another story. You won't be able to get a towel into all of the tubes to dry things by hand, so we'll have to rely on a couple other methods to keep the water off of the exposed metal. First and most important is air-drying. When the bike gets wet, take out the seatpost and flip the bike upside-down. Any standing water will pour out. Now filp the bike back right-side-up, leave the seatpost out, and let the inside air-dry overnight. In the morning, reinstall the seatpost and ride to work on a dry bike!
Second, and a little more complicated, is to keep water out of the frame in the first place. While you may not see them, there are small gaps between the seatpost and the frame at the seatpost clamp. This is where nearly all of the water enters the frame. To keep water out, get a cheap tube of grease, take the seatpost out, and slather the inside of the seat tube with grease. When you reinstall the seatpost, the grease will create a seal between the frame and the seatpost, and, if done right, will keep out any water. If you're out in a downpour, you'll still want to air-dry just in case, but in drizzles and showers you won't need to worry.
The final method is creating a protective layer on top of the steel, similar to what the paint does for the outside of the bike. There are two main products that do this: JP Weigle's Frame Saver (made for bikes), and Boeing's Boeshield (made for aircraft, but works just great and is even sold in many bike shops). To use these yourself, you must be comfortable with removing and reinstalling the bike's seatpost, bottom bracket, and as much of the headset as you can without pressing out the cups. A lot of that is beyond the average home mechanic and requires special tooling, so if you want your frame frame-savered, you'll probably want to pay a shop mechanic to do it for you. Using Frame Saver of Boeshield will prevent rust even if there is standing water in your frame, but it is not perfect and is no excuse for not actively taking care of your bike, so you should still grease the seat tube to prevent water entry and air-dry if you think water got in. If you take care of your bike with the other two methods, Frame Saver really isn't necessary. I don't use it, and I've been rust free for a year after riding in the rain and snow and whatnot.
If you already have rust, dilute oxalic acid is the preferred rust reducer. It'll chemically react with the rust to turn it into something nonreactive and ceramic-like (I forget exactly what), while not messing up your paint or decals. It can be found at most hardware stores as "wood bleach".
There's no way to attach a rear rack to the bike just because of how small the bike is and how it's set up. My DH tried to attach a rear rack last night, but the seat got in the way, and also there was no place to put a rear light. So it looks like I can have lights OR cargo, but not both.
There are very few racks that fit perfectly on any bike. The seat being in the way is one thing that may not be avoidable, but if one of the connecting arms doesn't go to quite the right place to be bolted onto the frame, don't be afraid to bend it to where it needs to go.