I cannot speak for the road bike Plus tires, but I can unfortunately say that the Conti Cyclocross Plus, despite being very heavy, is nowhere near puncture proof. Goathead thorns go right through the casing.
Installed Mr Tuffy liners, no problems since. The ultra-light road liner is 30g (per, 60g total front & rear) which is not that bad. They're also under $20 a set, which leaves a lot of budget to use any mid-grade tire or the OP might keep using the tires he already has.
The ultra-tough tires I know of also happen to be premium models and I doubt you can get a set for $70 or less, even on ebay or buying from the UK. (Schwalbe Durano Plus or Ultremo DD.)
Nails and large pieces of glass will puncture any road tire with air in it. So in the conditions the OP described, nothing is going to be 100% effective.