Try avoiding wall plates and any port extenders, etc. If you can run a straight cable with pass through wall plates you'll have a better chance but 35' is still a long run. Try this with your current cable if possible. If that doesn't work here are a few solutions.
1. Use a balun, either cat5 or fibre (the latter preferred but pricey, for longer distances definitely necessary but your run is not that long and cat5 would be fine). I recommend ones by Ethereal, Spectrum Electronic Solutions or Binary. For fibre Neopro integrator.
2. Use the current cable and try a signal restorer. I recommend the ones by Ethereal or Spectrum Electronic Solutions. This is a good option if it is difficult to run a cat5 or fibre optic cable.
3. Replace the current cable and don't use the wall plates as mentioned before. This is probably the least expensive solution but expect to replace it again in the future with one of the other solutions if you ever upgrade to 3D, Higher res. PJ, deep color sources, etc.