In your example, you'll want to flip your line as addresses typically ascend in the direction of the line itself. Your range would 592- 608 on the right side (after flipping) and 593- 605 on the left side. My personal preference is to have ranges start and stop with numbers ending 0 or 5: so I would range Peach as 590-610 and 591-609.
If you have access to the grid your city is based on that will help; Somewhere, there is a coordinate value of Peach as well as Cottonwood. In the 15 years I did gis for 9-1-1 dispatch I made sure the address ranges went through the intersections. In other words if Cottonwood has a value of 550, I would range Peach as 550-610 and 551-609. That way when a UPS truck collides with a Fedex truck in the intersection, you can address the wreck as 550 Peach. Remember, even though nobody lives in an intersection, all sorts of things happen at intersections.
I suspect the addresses you have plotted are from the Master Street Address Guide (MSAG) which isn't a bad way to start, but again, my personal preference is to range streets by the underlying grid. All the MSAG provides is an address where the local phone company provides dial tone. If I were to use the msag to range the street I live on, my address wouldn't show up since I don't have a land line any more and haven't for years.
That should just about do it....