I prefer used bikes myself. I have bought hundreds of bikes, many on C/L (many elsewhere too). Bought two bikes this week off C/L. Deals on C/L tend to be two types: 1) bikes that need a lot of work and 2) bikes where the seller just priced them too low. The bikes in the second group go fast, lightning fast, in most markets, as there are always people looking for deals. In the first category, bikes can last longer, as most buyers want bikes ready to ride. The two I bought this week had been on C/L for a while, bought one for $30, the other for $50. Neither was my size, so that adds a wrinkle in your search.

The other type of bike that will fit within your budget, and meet the needs you describe, is the old standard: rigid frame mtb. Yes, they are not as fast as racing bikes. But they provide tremendous value for the $$. Around here, you can get a really nice one, ready to go, for $125 to $150. Similar quality and condition racing bikes here go for double or triple.

That $30 bike above is a dud.

Spending less and getting more = looking harder and doing rehab work yourself. And the best deal around here are usually word of mouth or garage sales. Unfortunately, to find a decent bike at a garage sale takes going to a lot of sales, burning through a lot of gas. Your search is complicated by needing a specific size. In my case, any deal, any size, is OK.

Another choice is to find a few deals NOT your size, clean them up, air the tires, resell, and use the proceeds to dramatically increase your bike budget.