Are there recommended sites/activities for these locations in June? Thanks in advance.

I can recommend some strip clubs and love motels.
OH, gee, that isn't what you wanted? Then why don't you start by telling us what you DO like?
Do some homework, consult a paper or online guidebook and come back with specific questions.

Note: June is a month of "country" festivals all over Brazil, called "festas juninhas", that feature music and traditional costumed dance, the major one being São Joao on the 24th.

I have researched Lonely Planet, Trip Advisor as well as several on line sites and YouTube. We have several days of commitments in each location and was hopeful to hear about things that are favorites of locals that perhaps don't get a lot of publicity, such as the Palao Musica is Barcelona. We are urbanites who enjoy hiking and taking in the local flavor, not much for just sitting on a beach although we do like to walk them. There is the weekend market and the tower to climb in Campinas and the dune buggy experience in Natal but I haven't seen much else which is why I inquired here. Strip clubs and love motels are not the usual recommendations on this site.

Maybe not, but love motels are as local as it gets in Brazil. Everybody and their grandmother has been to one at one time, given the lack of privacy in the typical crowded, multi-generational Brazilian home. Some are actually quite luxurious.

If you paid attention to the blurb on LP, you might remember the word "bland" used in reference to Natal. There's a fort, a theater, a bridge, all pretty minor as tourism goes, and the dunes at Ponta Negra where the package tourists stay, but passing through to the beaches of Pipa or Genipabu is probably the best reason to be in Natal.
Locals go to the beach and drink beer, play foot vollei, etc. As mentioned, in June, they go to the festas juninhas to drink and eat corn-based foods, and watch the dancing.
You will probably not be doing much hiking up there in the humidity, high temps and probable rain.

Campinas is another not too exciting city, mostly known to tourists for its airport, now used as an alternative to the busy one in Guarulhos serving São Paulo. There are some minor buildings and a lake, and the usual provisions for locals to drink and dine. The logical visit would be to sprawling, congested, fairly nearby São Paulo itself, but it is not a particularly touristic city either, and one that gives up any gems only with time and the native language.
There are some mountain areas not too far from Campinhas, good for overnight stays, where you might find some hiking, if the weather cooperates.

thank you for your help

In Campinas, a top restaurant choice is Theo Medeiros. For more local flavor, with choro and samba many evenings, try Emporio Nono.

Why don't you explain to the OP what choro is? They probably aren't familiar with it.
Is the samba you refer to samba enredo or samba gafiera? Or something else? Please explain to the OP the difference.

Here is the website for Emporio do Nono: http://emporiodonono.com.br/a-casa.