It'll be a struggle to find a hotel for under $100, including taxes/fees, in pretty much any of those downtown areas near the train station. If you're lucky, you may be able to get some deals on the opaque booking sites like Priceline and Hotwire, but they're non-refundable and you don't know the exact hotel until you book. As such, I'd make sure to only go for it if they're significantly cheaper than comparable hotels on regular booking sites, and I wouldn't dip down below a 3-star property. I'd also make sure that whatever mechanism they have for showing the average user review shows quality reviews (7/10 would be bare minimum, 8/10 is better.) Also be careful on the boundaries of the "neighborhood", as sometimes they'll be fairly large and have one or two hotels right on the edge of the boundaries that are more difficult to get to.
Another good option in many of those cities, if you're okay with compromising on travel time, is to book an airport hotel. In Chicago and Seattle for sure (I'm not sure about New Orleans or San Diego) the local rail transit goes to the airport, where you can access many hotels' airport shuttles. You'd likely have a long-ish commute time into the downtown area, but it'll be much easier to find a hotel in the $80-$85 range (which will result in at or below $100 after taxes/fees in most cities.)