There are many different solutions depending on how much you'd want to spend, how much DIY you want to do, and what exactly you are trying to achieve. I would not try to directly switch this load: inductive loads generate voltage spikes when switching them, which can damage electronics and such that are not designed to handle it. They also have high in-rush startup currents which are beyond the normal ratings.
On the cheapest, very DIY side, I would use a definite purpose contactor for each pump (~$20, give or take), and run some smaller (14/2) wires inside to a switch.
The benefit here is the wiring going inside is low-current, and so you can use a regular lightswitch, count-down timer or programmable timer.
Insteon is another way to go. Insteon is a power-line carrier signal, which means it runs over your existing wiring. They have a high-current relay module which would be used to control your pumps.
You can then control from any Insteon-capable controller: there are a number of keypads, light switches, timers, home automation controllers, key chain remotes, motion sensors, and PC interfaces. The benefit of this is your "remote" can be anywhere (and you can even have multiple remotes). For example, here is a 6-button keypad that would work:
You could have a timer/controller set up to run your pump at certain times of day, or have a motion sensor that runs it if someone is in the pool area, or a button to manually trigger it. With a controller you can do very complex things, like have it run if someone is in the pool area for more than 30 minutes, or if it's manually triggered, and also ensure that regardless of that, it runs for at least 4 hours every day. Heck you can go crazy fancy and have it run more after it rains, or after it's particularly windy (to get leaves out).
I've linked to smarthome.com (who is the manufacturer of Insteon) but you can also buy their stuff from many other places, including Amazon.com.
There are equivalents to Insteon in other technologies (both wireless and power-line carrier) such as Z-Wave, UPB and X10 (though X10 is very poor technology, I would recommend staying with something newer).
You can also combine. For example, there is a UPB dry-contact output which you could wire up to a definite purpose contactor.
I don't know if this is really an answer, but I've had a Coleman 6 person spa in service for about 25 years. Yes, 25 years! I have never had the same problems you are experiencing. We learned early on that any clothing (bathing suits etc) must be very clean and rinsed extremely well before going into the spa. Clothes cleaning detergents left in the fabrics release into the pool water and cause foam quickly. The clarifier is good at flocking these soaps, but they do tend to gum up the works quickly. We never lost a lot of pressure however, just had to clean the filter. We also made a habit of completely changing the water at least twice a year. After draining as much as we could via the drain hose, we used a wet vac to get as much out of any low spots. We then put in just enough water to cover the intake close to the bottom and jogged the pump to cycle clean water through the system and get the dirty water out. Again, drain and vacuum and repeat this process until the water looks clear. We do this with the filter out. This will clean out a lot of crap. About once a year just before we do a drain and complete cleaning, we treat the tub with anti-scale cleanser. This cleans the internal pipes of built up scale and solids. In your case, you may want to do this anti-scale treatment for a few days with the circulator on low and the filter out to purge the internal system. Then go ahead and change the water.
Another test you can do is to see how well the pump pushes water. When you have the small amount in the bottom, just above the intake, jets exposed, turn the jets to full, air off and see if the pump will shoot the water out of the jets with extreme force. Mine will shot water 20 feet away!!!! lol. If you don't have really good pressure, you may have a warn pump impeller. They can be damaged over time by chlorine based sanitizer, I never use chlorine, just bromine based shock etc.
The other item to check is if you have the right filter for your unit. Filters come in so many styles and densities. You may need to find one that allows better flow.
Assuming you find the root cause of your problem, good housekeeping, keeping as much contaminates as possible out of your tub will help. Very clean bathing suits or none at all,(yahoo) no body cosmetics such as body oils, moisturizers, perfumes etc. Keep chemical treatments to a minimum. Do a quick check weekly for PH, Total alkalinity, and bromine level. Adjust them sparingly so not to overdo any chemicals. Once you get everything under control, maintaining it is very easy and quick. Good Luck
Best Answer
In my experience (since the prior owner is not being helpful) there's probably an outlet someplace that this just plugs into. Should be outside, have an "in use cover," and should be GFCI protected (preferably by a GFCI located inside,) but none of those may be true, or all of them may be.
And in all likelihood, the wiring is not remotely up to code - All too common to see an extension cord just barely below the surface of the grass, or running through a flowerbed. So, grab a trowel and follow that wire. I'll be shocked if you actually need a shovel (implying proper burial depth, though wrong type of wire is still a strong possibility.)