I'm 59 years old. What are my options for getting rid of double chin and loose neck skin without a face or neck lift? (photos)

Skin laxity under the chin, also known as “turkey neck,” is a common complaint among men and women alike. This issue usually consists of some submental fat that has accumulated under the chin, as well as loose skin that exacerbates the problem. Because the problem is twofold, the solution also requires combination therapy that can be done in one procedure. 
Submental fat can be eliminated using a gentle lipo technique that uses tumescent fluid to minimize trauma to surrounding tissue. The procedure can be performed using local anesthetic and downtime is generally minimal. Immediately following liposuction, ThermiTight is used to reverse much of that skin laxity and create a smoother contour under the chin by delivering deep heat with a radiofrequency probe. I recommend finding a board-certified dermatologist who can perform this treatment to your precise needs for the best possible result. 

I appreciate the photos you included with your question. Based on what I am seeing, I am not sure a nonsurgical treatment will work best for you. While chin liposuction will eliminate the submental fat underneath the chin, the removal of the fat is likely to make the skin laxity in the area even more pronounced. It is very possible you will need a lower facelift and possible neck lift to achieve the results you want. These procedures can be performed at the same time for most patients, giving you full results with just one round of anesthesia and one recovery process. Make sure you work with a board-certified plastic surgeon to ensure your procedure is both and safe and successful. 

Liposuction is an excellent option for removing stubborn submental fat under the chin and creating a more defined jawline. However, if there is skin laxity, that can be exacerbated by performing liposuction alone. I would probably recommend a combination of liposuction and nonsurgical skin tightening to produce the best possible results for you. 
I use the Awake lipo technique, which can be done using local anesthesia and only requires a few days of recovery. Incisions to insert the small tubes for suctioning out fat do not leave much visible scarring behind. ThermiTight is a skin tightening procedure that delivers radiofrequency energy deep into the dermal structure via small probes. As the tissue is heated, it constricts. Collagen production also increases, creating smoother, firmer skin from the inside out. Always work with a doctor who is board certified and experienced in these procedures to ensure an optimal outcome.