I would say that if the partial hurts where it clamps into your mouth, which I assume would be on a permanent tooth, that also should NOT be and it needs to be adjusted before the tooth it clamps to is ruined.
Lastly, I would say that if you are in bad enough pain that you are asking about pain medication, you definitely need to go back to your dentist and don't be afraid to tell him what is wrong and explain that it must be fixed.
So sorry to hear how much pain you have been in. Your adjusting to dentures journey is just beginning; even for experienced denture wearers, new dentures can be a challenge. Don't worry; you're not alone! Some 36 million Americans have dentures too. Many first-time denture wearers say that eating soft foods that are gentle on your gums/teeth makes this new experience easier.
The first few weeks you mouth is adjusting to the new dentures; you will likely experience increased salivation and sore spots in your mouth from the dentures. As others have stated, rinsing your mouth with warm salt water might help. If soreness persists, return to your dentist for an adjustment. Expect a longer denture adjustment and healing time if you recently had teeth extracted or are a full-plate wearer. I know it seems like forever right now, while you are in so much pain, but some day you will be wearing a smile and not giving it a second thought! Hang in there and let us know what the dentist says. Warm holiday wishes to you and yours.