Hi..
Welcome to HEALTHCARE MAGIC..
I have gone through your query and can understand your concern.
As per your complain in case if you have got a 
sore throat and 
cough after using a new denture cleanser there is a possibility that you have not cleaned the denture thoroughly with water after placing in the cleanser and there has been chemical 
injury to the oral and throat mucosa with the components of cleanser and it has lead to the symptoms..
You should thoroughly wash the retainer with water before you place it in mouth..
In case if the problem persist you can change the cleanser containing a different composition after consulting your 
dentist..
As of now you can do warm saline gargles and steam inhalations to relieve sore throat and cough..
For burning you can gargle with a numbing mouthwash containing Lignocaine and also suck ice pops..
Avoid spicy food and maintain a good 
oral hygiene..
Hope this information helps..
Thanks and regards.
Dr.Honey Nandwani Arora..