I recommend you see a neurologist to determine the cause of your facial droop. It sounds unlikely that it could be from the Botox one month later. Please let me know what the neurologist suggests.
Facial paralysis 4 weeks after Botox is most likely Bell's Palsy and not due to Botox
Bell's Palsy is when all or a portion of one's facial nerve doesn't function properly and there is NO OTHER CAUSE FOUND. Not all facial palsy is Bell's. You need to see an ENT doctor to get properly treated and to rule out other causes such as: tumors, autoimmune disease, zoster, systemic disease, trauma, botox. A hearing test and balance test might be needed and nerve testing too. Botox as a cause, if it was Botox you got, would lead to problems within 10 days. It would be very unlikely that is was due to Botox but I'll never say never.
Botox problem or Bell's Palsy
You should see a neurologist for confirmation on whether or not it's Bell's Palsy. A neurologist can appropriately test and diagnosis this. A problem with Botox would have happened more quickly than 4 weeks out, so based on that part alone, I'd say it's Bell's Palsy. But have an appropriate exam to find out for certain.
"This answer has been solicited without seeing this patient and
cannot be held as true medical advice, but only opinion. Seek in-person
treatment with a trained medical professional for appropriate care."
Drooping lip after Botox and filler
If there are any concerns 1 month post-Botox, a follow up visit is in order. Bell's palsy can occur at any time (and often without known cause), and a PCP or neurologist can help sort out if this condition is present. Outside of Bell's Palsy, if the depressor labii inferioris is inadvertantly weakened with chin botox injections, the weakened side will rest higher upon smiling. That is because the muscle that normally pulls down that side of the lip is weakened. The way to correct this is to inject the depressor labii inferioris on the unaffected side so both muscles are weakened. Alternatively, a patient can wait it out to see if the symptoms improve with time.
if your doctor thinks it is Bells Palsy then you should be seen by a neurologist to confirm and rule out that this is from your Botox treatment.