If the RCA wires are loose, you may be able to use the old ones to pull new wires through the same space. It would not be a bad idea to pull some string at the same time. (Mason's line is one reasonable option for a decent string to pull; it is fairly sturdy and inexpensive and lasts a long time.)
If you do run wires, be sure to use decent quality HDMI cables for relatively long runs. HDMI is not a very well-designed system for long cables. It was developed from DVI, which was intended to connect desktop computers to monitors and so intended for fairly short distances. Without going into technical details, suffice to say that especially for longer cable lengths poor quality cables may not work reliably.