Are you checking the correct pins? The DB-9 has Request To Send as pin 7 and Clear To Send as pin 8. Are you using an oscilloscope? RTS is the signal you can control. CTS is a response, say from a modem, but you can loop the two for testing, where if you assert RTS, CTS should go high. I do not recall a software setting to disable these.
Try Terminal by Bray. I have not used RealTerm in about 4 years, but turn on hardware flow control if it is an option. I had a Dell 800 and the monitor went bung (a common problem) shortly after the warranty ran out. You are lucky yours is still going.