Towing add a lot of extra load to the tow vehicle, and it can expose characteristics that you may not notice when driving solo. This is one of the reasons why it is so important to ensure your tow vehicle and trailer are maintained in good condition.
As the problem seems to be speed related immediately suggests to me it could be insufficient nose load. The static nose load you set by adjusting the location of any load in in the caravan should counter and exceed the towing aerodynamics of the caravan which generally produce a lifting torque on the hitch which is very much speed dependant.
It might be worth getting the cars tow bar checked for loose fixings.
Remember that many cars should have increased tyre pressures when towing, check your owners manual for details. Tyres are of course very important, but often if stood for long periods heavily loaded (and caravans usually are) they tyres can stiffen and develop a flat spot which especially with older tyres may become a permanent problem. But that would be noticeable at all speeds.
You tell us the caravan has been sited for the last two years, in which case it's unlikely to have had a "full service" on its running gear. mindful of the fact the caravan is now 17 years old, it's far more likely to have deeper issues. That's one of the reasons why I and others strongly recommend having an independent pre purchase inspection carried out by an Approved Workshop Scheme engineer to establish the condition of the caravan.
I strongly advise you have the caravans running gear checked, and also look at the advice about where to load items into the caravan to improve towing handling.
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