Welcome to TFP!
CYA isn't just for protecting chlorine from breakdown by the UV rays in sunlight. CYA binds to chlorine significantly reducing the "active" chlorine concentration that does the actual disinfection and oxidation. This is not a bad thing since it takes very low levels of active chlorine to kill most pathogens. So you do want at least some CYA in the water or else the chlorine will be too strong and will oxidize swimsuits, skin and hair too quickly and will also tend to outgas faster which may be what you are seeing.
I suggest you start off using Dichlor as your source of chlorine after an initial fill for a cumulative Free Chlorine (FC) dose of around 33 ppm. That will get you to around 30 ppm CYA. If you use the tub every day and add around 2 teaspoons of Dichlor after every soak in 350 gallons, then this is around 8 days of Dichlor usage. After that, switch to using bleach -- Clorox Regular unscented 6% bleach.
If you still find your chlorine usage to be high after doing the above, then shock the tub with bleach since you may have chemicals that need to be oxidized from the fill water. If your tub is new, then you'll need to replace the water much sooner on that first refill since there's often a lot of oil and contaminants in a new tub.
You say you are using HTH for your tub. Can you be more specific about the HTH sanitizing product you are using? Is it Dichlor? Is it bromine tabs? Is it Cal-Hypo? I can see that HTH Spa Shock is Cal-Hypo and it appears that HTH Spa Chlorinating Granules are also Cal-Hypo. It would not be good to use too much Cal-Hypo in a spa. If the Calcium Hardness (CH) gets too high, you can end up with scale. If you have a good test kit, such as the
TF100 from tftestkits.com, then you can check your CH level and stop using the Cal-Hypo when the CH gets to around 120-150 ppm. So you'd start with some Dichlor, then use Cal-Hypo, then switch to bleach -- if you really want to use the Cal-Hypo (otherwise, just go from Dichlor to bleach).
Richard