There are several different possibilities here, and definitely an allergic or sensitivity reaction is one of them. Often people with sensitive skin will have a contact reaction to new detergents or chemicals. This can lead to areas of redness and itching as you describe. The best way to know if this is what is going on is to eliminate the potential source of irritation (e.g, the cleaning service or the new detergent) and see if things improve. Keeping the skin well moisturized in the interim and applying some over the counter steroid cream for itch should help as well.
"Heat rash" is another common complaint and occurs when excessive sweating leads to temporary plugging of sweat glands. Small red bumps appear which may be itchy. Heat rash generally responds to just keeping the skin clean and dry.
Folliculitis is an infection of the hair shafts which leads to redness around the base of the small hairs in the skin, sometimes with a small pimple. In addition to heat and friction, like with heat rash, other things that predispose to folliculitis include waxing and shaving the skin. Again, it tends to respond to good skin hygiene.
If the condition worsens or doesn't respond to these simple measure, go see your primary care doctor
, who can diagnose your condition formally and suggest an effective treatment.