I am not sure at all to define as 'damp' the weather in Lima in the 'winter' season. We have high humidity but that's it. And as you must know, humidity plays a role in the sensation people can perceive, the 'feels like', both in summer, making you sweat like a pig and making your skin sticky and in winter making you feel a cooler weather than it actually is.
It can get a little foggy sometimes, but we have virtually no rain at all. All we get is some sort of a weak drizzle, sometimes it can wet the streets, but never it goes further.
July, August, September, October are the coolest months, from November January it begings to get hotter, from February to June you can get easily bad sunburns if you don't wear sunscreen. It is mostly a dry hot weather all the year.
Here is Lima most people don't know what heavy rain, storms, tornadoes, hurricanes are, only a few who have travelled can answer properly. Technically Lima is located in a desert...
Like @Dave, I prefer 100 times this weather. Specially when it is not 'summer' I don't like that much the sun of Lima. Where I come from we have like 100 all year long.
And like @ironcheffchris said.. Tourist from northern USA and Europe are in flip-flops when the temperature is higher than 10°c lol... It is their kind of summer, I was able to get that impression in a trip to Cusco, we were at 11°c.