Hi Downinahole, welcome to The Mattress Underground, Glad to have you here 
Sorry to hear you are having issues with your old memory foam mattress. I agree, unless you know all the specifications you should avoid any mattress whose components do not meet Mattress Durability guidelines, which it looks like you have seen. This is a good place to start, but which types of mattress are you considering? You have had a poor experience with your foam mattress, are you considering another, or only a latex or hybrid one? What materials do you prefer?
Even though the Memory foam is “cheap and sagging” and clearly a cause of pains and spinal misalignment you may still want to check that the foundation underneath does not sag under the weight of the mattress and people sleeping on it. Does the storage bed frame have center support to the floor? I’d Ask hubby to stand on different areas of the frame to see there is any give or bending, to make sure the foundation is not part of the problem as well and ensure that this foundation would be appropriate for a new mattress.
We do not usually recommend any mattresses or companies, as purchasing a new mattress is a personal choice, every person is different, and should be based on their specific needs and preferences. So we prefer that you go through the process starting with the mattress shopping tutorial and narrow it down to a few good mattresses for which you find the mattress specifications you need to know so that we can make comments that would better help orient you.
$5,000 is a very generous budget given you chose correctly in terms of PPP (Posture and Alignment, Pressure Relief, and Personal Preferences) using the Durability guidelines, such mattress can last you a long time; After all a mattress is as good as the materials inside it. That being said, because of several the criteria you listed (“sleeping hot/cold issues, no chemicals and skin sensitivities”) CozyPure one of our Trusted members comes to mind, they are located several hours away from you in Norfolk, VA, but they offer great quality, excellent service, and offer natural organic products that you may want to take a look at and try. A shop visit and local testing is usually the best way to decide on the type of mattress you can do well with. CozyPure has a dedicated forum page here
Thanks for providing your stats; since there is a large weight differential between the two sleeping partners, you may want to look into a split side by side mattress configuration - this would allow you to choose a mattress suited to each of your personal preferences. You also might want to consider an adjustable base for your mattress as you seem to need some head elevation for your “breathing/chest heaviness issues”
Since you are a ‘hot/cold’ sleeper, temperature regulation is important. Wool is known for its unique temperature regulating abilities, and is a natural material which keeps you warn in the winter and cool during summer. This would certainly benefit you when used in the cover or quilting in top layer. It has a structure that has a more porous inner core which can absorb large amounts of moisture without the moisture coming into contact with the skin. This moisture which is held inside the wool fiber itself can then evaporate in a more gradual process which helps to regulate temperature much better. Its’ also a natural fire retardant. Another material you may want to consider is latex which is the most durable of all types of foams, and a breathable material which would help you keep a neutral temperature range. You can see more about latex benefits in this post
Lower back pain, especially in the morning, could be indicative of improper spinal alignment. Since you have some pre-existing conditions, I checked with Phoenix, who mentioned recently buying a saddle stool to help with posture while sitting at the computer desk for long periods. You may consider looking at something similar to help with posture during work hours.
Head elevation can be achieved with pillows (even wedge pillows) or with an adjustable base which has separate side of bed adjustment, allowing elevation of the head which can alleviate breathing issues. Another of our trusted members, as above, Richmond Bedding in Richmond VA carries an adjustable bed you can look at.
The happiest consumer is a knowledgeable one!
I hope this helps,
Basilio