What is the Best Solution for Eyes Losing Their Shape After Weight Loss?

This does not sound healthy as you are periodically dieting and probably dehydrating yourself to the point where your eyes change in order to compete. Never mind your eyes think of the damage you are doing to the rest of your body with this behavior. Furthermore, if you put in a filler what are you going to do when you regain the weight and water so you no longer need the filler but it is still there?

My response to your question/post does not represent formal medical advice or constitute a doctor patient relationship. You need to consult with i.e. personally see a board certified plastic surgeon in order to receive a formal evaluation and develop a doctor patient relationship.

Fillers would a good option for you during your "lean days". What you have to realize that when you gain weight again, this could alter your "ideal look". The fillers last for 6-12 months and if you don't like the look during your heavier days, you may need to get the product dissolved with hyaluronidase.

Juvederm is a cohesive filler product.  This is a rather counter intuitive descriptive name for the "smooth" character of this product.  This is helpful for inexperienced injectors treating noncritical areas of the face because lumps and bumps create by the treatment.  However in the periocular area this tendency of the product to spread out created some unwanted post treatment swelling that can be very difficult to resolve.  Restylane has proven to be a much better product for this type of off label treatment.

The eyelid hollows that can occur with weight loss can be filled with fillers like Juvederm. This must be done conservatively so u do not get bumps. 

Loss volume causes loss of support as you have noticed. In order to reverse this, you will have to put back the volume and fillers, synthetic or natural such as your own fat can do this.