Hello and hope you are doing well.
The rash appears to be Hives, which develop in response to an allergic reaction, by the release of histamine. The allergen can be medications, food, extremes of temperature, exercise and viral infections to name a few. So, therapy lies in avoiding the allergen and anti-histamines. If the present anti histamines are not very effective you can consult your doctor for long acting ones.
Hope this helped and do keep us posted.
Rashes that affect the palms and soles could be:
Atopic dermatitis
Contact dermatitis
Drug eruption
Erythema multiforme
HIV acute exanthem
Kawasaki disease
Lichen planus
Meningococcemia
Psoriasis
Rocky Mountain spotted fever
Rubella
Scabies
Secondary syphilis
Staphylococcal scalded skin syndrome
Stevens-Johnson syndrome
Tinea corporis
Toxic epidermal necrolysis
Toxic shock syndrome
Urticaria
You included some history in your post. Clearly you are aware that diagnosing a rash requires a detailed history. Have you been around any sick people? Are you exposed to chemicals at your work or any hobbies in which you engage? Do you live in an apartment building where an exterminator may have sprayed? Don't forget that this is the time of year when you might have eaten some new food. Did you get new clothes? At first blush I am tempted to say you have atopic dermititis, but I will not since I do not have a complete history, nor can I see you. You should make an appointment with your primary care provider.