Botox can, and for some does, cause dry, bloodshot eyes. The FDA agrees. Doctors please don't delete this; don't be defensive.

There are various side effects that pertain to Botox. As there are various reasons for dry/ red/ watery eyes. Without a full medical history and in person assessment it is difficult to give an accurate judgement. Yes, it is a possibility your condition is caused by Botox, in which case it will wear off in approx 3 months. Although, you may require a consultation with an optometrist.   

If the Botox is injected too close to the lacrimal gland, it can decrease tear production and cause dry eyes.  If Botox is injected in significant quantities to the muscle under the
lower eyelid, you can be left with some difficulty closing the eyes, which can also contribute to dry eyes. This is because the muscle is no longer able
to contract or tighten as before the treatment.  It's not an allergy -- it's an anatomical issue.  It's rare that this stuff occurs, but you can lubricate your eyes with artificial tears until the Botox wears off.  I feel only "core" injectors should perform cosmetic injectables (facial
plastic surgeons, plastic surgeons, dermatologists, or occuloplastic
surgeons). This is because we all have a keen understanding of facial
anatomy and the effects of injecting Botox/ Xeomin/ Dysport and fillers in different
regions of the face. I wish you well!

Botox injections around the eyes can certainly cause the eyes to become red, dry.  They weaken the muscles that cause wrinkles.  Wrinkles around the eyes are caused by the orbicularis muscles that also close the eyes.  Therefore if they are weakened by Botox they will not blink or close over the eyes as well and can cause dryness, redness in some patients.  In fact, under rare circumstances we will inject the main lacrimal gland area of a patient with "Crocodile" tears from misdirected 7th nerve fibers to make the gland produce less tears.  Sometimes injections over the lacrimal gland area for wrinkle improvement may affect this tear gland as well.  Usually, however with the use of quality artificial tears or lubricants this situation of botox worsened dryness is quite well tolerated.  I practice in Colorado and most adults have some degree of dry eyes, and I still inject botox injections to the crows feet areas without significant concern.  I will place less to the lower eyelid areas in severe dry eye patients.  I hope this helps. I'm sorry if you have had red, dry eyes after botox.  The effect should wear off as the botulinum effect wears off in a few months.  Best Regards.