Both are great, neither have a lot of traffic compared to where I live.
The main Alps passes that link larger towns or are main routes across the mountain range will be busier, the Stelvio and Grossglockner are always busy, but you might still want to take them in. I have done many Alps trips from July to September, never have I found the traffic unbearable, just the odd spot here and there, never enough to ruin a whole day or even half a day for that matter.
The Dolomites are possibly the worst for traffic if generalising areas, certainly the figure 8 loop around Arabba, but it is also one of the most spectacular days riding available in the Alps, head a bit south of there away from the larger towns (holiday resorts) to passes such as the Manghen and you will find beautiful scenery, great loads and very few cars / buses / motorhomes.
If you can steer clear of any large lakes and the towns near them you will avoid the very worst traffic the Alps has to offer - but maybe you would like to see some of these lakes, IMO it is all about balance and if you want to ride the Stelvio, or see Lake Garda them do so, I am sure you will remember the views and experience of being there long after the memories of having a few tin boxes in your way has faded.
I have not ridden in the Picos and only had one trip to the Pyrenees, they generally are lighter on traffic, but there are also less iconic routes and passes, if you have the time I would suggest trying to ride in both if you can, to link them up head across the southern reaches of France as generally it is very nice, the North is more congested and lacks the scenery, roads and climate of the South.
Grab a copy of the book "Motorcycle Journeys through the Alps and beyond", it is written by an American so you should understand it better than I did, it has routes in the Alps, Pyrenees and Pico's - although mainly the Alps. It should give you some great ideas of the bits you might like to work into your trip.
My top tip would be RDGA - if you can find the time to stop for a couple of days in a couple of places to take in some of the routes along the way (in your fellow countryman's book) all the better. The 7 pass loop around Mont Blanc is fantastic and any circular route out from Barcelonnette is well worth doing.
Personally I get a lot of enjoyment out of researching and planning my trips, I hope you find equal joy in doing this as your trip approaches, as a good American friend of mine always tells me "if you do not have a plan then you have nothing to deviate from" or my favourite quote from another great American "planning is everything, the plan is nothing"
Have a great trip.
