Just took off EB Cobalt flats and put on TI flats. Yup, definitely more old school sounding. If both of these have similarities to rounds, I'd say Cobalt flats are closer to SS rounds but without the fret noise, and TI flats are closer to worn in nickel (or nickel plated) rounds. More of that flatwound thud on the TI flats, so I'm digging them more for that old school tone. Less brightness, more mids, warmer and woodier tone. EB cobalt flats are a different type of awesome, clearer and brighter tone, but muscular.
I honestly remember TI flats being more floppy, but it's been like 4 years since I've tried them. My setup is already fitting for TIs because my A string saddle is always the highest (slightly higher than E, then the D is similar height to E and G is the lowest - fits TIs perfectly). So no fret buzz, and what's surprising, I didn't even have to adjust the truss rod one bit. Maybe I'll have to loosen it just a tad once the neck settles, but so far, really great for standard tuning. Helps that Stingray pickup is closer to the bridge than a P pickup, so playing them over the pickup is awesome on the ray. Also helps that I usually play rounds more aggressively than flats, so it all evens out.
I'd definitely recommend trying out both EB Cobalt flats and TI flats on a Stingray.