Your dentist must have asked you not to remove the dentures ,so that the extraction sockets are not exposed to the oral cavity
,to prevent any infection.
However,if you are having discomfort because of the denture,you can remove them for some time.You donot have to wear them all the while.
Make sure that you maintain good oral hygiene.
Ideally dentures are removed at night ,before sleeping.
However,since you have just got some teeth removed,your dentist must have asked you not to remove it.The dentures would act as a physical barrier and would ,to an extent protect the extraction sockets.
In my opinion,it is not necessary to wear the dentures all the time.You can remove them for some time and wear them again later,if you are having discomfort.
The extraction sockets generally heal in 3-4 days.Since you are having Diabetes,wound healing would be delayed and it could take a longer period of time for the extraction sockets to heal.
Maintaining a good oral hygiene is necessary.
You can use a antibacterial mouthwash,
Warm salt water mouthrinses would be helpful,
It is also important to keep the dentures clean.
Your dentist would be able to tell you ,after an examination, if you are having a dry socket or any other complication.
There is also a possibility that the extraction sockets are getting infected.Hence an antibiotic course would be necessary ,in my opinion.