When I talk my bottom row of teeth appears to slant or slope downwards, what can be done to fix this?

Hi,
Sounds like there is misalignment of your upper and lower jaw, which is affecting the way your teeth line up. Best approach would be to find someone who practices physiologic based orthodontics or dentistry. They can determine if in your case ortho or veneers would fix the problem. A photo would be helpful. 

Hi:
Unfortunately without photo and X-ray it is difficult to give you accurate advice.  Please post photo and X-ray of the affected area.
Best,
Dr. Maddahi

I would have to suggest that your orthodontist is giving you correct advice on this as s/he has considered the amount of bone already in your jaw, the amount of destruction he can see for radiographs and the chances of more problems occurring.
Given that orthodontics is out, how will veneers or bonding work?
Well, best thing to do is find a cosmetic dentist who does either digital smile design or can work with a lab to create a mockup for you so that you can see how either of these two techniques can give you the result you want. If you are not happy, they can design an alternative look.
Hope this helped and good luck
Kind regards
Thomas Dey, BDS, MSc