First of all, the fact that your year old Beardie is "only" pooping once a week is not a problem at all, as they get older they naturally do poop less and less often, with some adult beardies pooping only once every 10 days to 2 weeks. There is absolutely no reason at all to try to make him poop more often than once a week, unless he becomes impacted, which your beardie obviously is not if there is water in his poop. So you don't need to do anything at all to try to get him to poop more often, because he is just simply getting older and this is completely normal and natural. There is no reason at all to get worried unless at least 12 days has gone past, and even up to 14 days is normal, unless you see him straining to go and he can't, which indicates an impaction. But your beardie is fine as far as his weekly bowel movements go.
That being said, his diet is not very good at all, and does need some tweaking. Adult dragons will usually start to switch themselves from eating a ton of live insects every single day and little or no greens/veggies to the exact opposite, a ton of greens/veggies every single day and less live insects. As such, it's extremely important that you replace all of the nutrition (protein, fat, carbs, vitamins, minerals, etc.) that he was getting when he was eating a ton of live insects every day with appropriate and very healthy greens and veggies that contain the same nutrition. There is absolutely NO kind of lettuce that is going to provide him with even a fraction of the nutrition he needs to survive, as already said, all lettuces are just nothing but water, that's it. They have very little nutritional value, and he will end up becoming malnourished if you keep feeding him lettuces. It is also the lettuce that is causing all of the water in his bowel movements. So you need to totally stop the lettuce and then scale back the carrots, as they don't contain much valuable nutrition either.
Instead, you need to give him a mix of fresh (always fresh, never ever give him any freeze-dried greens/veggies or bugs), healthy greens that will replace all of the nutrition he's going to be losing as a result of not eating nearly as many live insects.
The fresh greens you need to feed him include Collard Greens, Turnip Greens, Mustard Greens, Dandelion Greens, Endive, Escarole, Arugula/Rocket, Bok Choy, Pal Choy, and Swiss Chard. All of these are available very, very cheaply (as cheap as the lettuces) at any grocery store or Walmart.
The veggies that you need to be feeding him daily mixed into his salad include all types of Squash (Butternut, Acorn, Spaghetti, Yellow, and Green), all types of Bell Peppers (Red, Green, Yellow, Orange), Green Beans, different types of Peas and Pea Pods, Broccoli, Cauliflower, and then sparingly a little carrot. Limit any fruit to only 1 or 2 times a week at most, as it contains quite a bit of sugar, which is bad for Beardies, but once or twice a week a few pieces of fruit is okay, like apples, raspberries, blueberries, blackberries, peaches, strawberries, etc. NEVER FEED A BEARDED DRAGON ANY CITRUS FRUIT OR AVOCADO!
Now the other issue, as already mentioned, is that your tank temperatures are WAY TOO HOT, and this is going to result is not only dehydration but also heat exhaustion, AND THIS IS THE REASON HE IS ACTING LETHARGIC AND HAS NO ENERGY TO MOVE AROUND MUCH, HE'S FAR TOO HOT!
If you don't already have either a Digital Thermometer with a Probe on a Wire ($10 at any pet shop) or a Temperature Gun, you absolutely must get one, as YOU CANNOT MEASURE HIS BASKING SPOT TEMPERATURE WITH A STICK-ON THERMOMETER! You only listed 2 temperatures, a Basking Spot Temperature and a Cool Side Temperature, and usually that is an indication that the person is only using stick-on thermometers, If this is the case, then that means that the "Basking Spot Temperature" that you listed is not at all the Basking Spot Temperature, as this is a SURFACE TEMPERATURE and cannot be measured with a stick-on thermometer. What you're actually reading on that stick-on thermometer is the Hot Side Ambient (air) Temperature, which should only be between 88-93 degrees F maximum, so this is obviously a serious problem if that's the case.
BASKING SPOT SURFACE TEMPERATURE (Must be measure with a Digital Probe Thermometer or a Temp Gun; Probe must be placed directly on the basking spot/platform where the Dragon lays to bask, and be allowed to sit there for at least 20-30 minutes before the temperature is read): Between 105-110 degrees F maximum for a Dragon under a year old; Between 100-105 degrees F maximum for a Dragon over a year old; 110 DEGREES F IS THE MAXIMUM SURFACE TEMPERATURE THAT SHOULD BE ANYWHERE IN A DRAGON'S TANK.
HOT SIDE AMBIENT (air) TEMPERATURE (The air temperature of the Hot Side of the tank that is SURROUNDING THE BASKING SPOT/PLATFORM): Between 88-93 degrees F maxium.
COOL SIDE AMBIENT (air) TEMPERATURE (The air temperature of the Cool Side of the tank; This is very, very important, as your dragon absolutely must always have a place that is considerably cooler than the Hot Side of the tank to go and cool down in): Between 75-80 degrees F maximum.
*****Your Cool Side Air Temperature cannot go above 80 degrees, your's is far, far too hot, and if you are not using either a Gun or a Probe to measure that "Basking Temperature" as you listed it, then the Hot Side Air Temperature of his tank, which should be only between 88-93 degrees, is actually up into the Lethal Zone, as will be the real Basking Spot Surface Temperature, because it is going to naturally be much, much hotter than the Air Temperature surrounding it. This is a very serious problem that you must rectify immediately, and it's the sole reason that your dragon has no energy and is severely lethargic.
****Also, as already mentioned, you cannot keep a Bearded Dragon in a 20 Gallon tank, not only because it is way too small for a 7 month old Dragon space-wise, but because obviously it's extremely difficult to keep an adequate and appropriate Temperature Gradient inside a 20 Gallon Tank. At 6 months old he needed at least a 40-Gallon Breeder Tank, and at a year old and for the rest of his life they should be in an enclosure no smaller than 4'x2'x2'. So you absolutely must get him at the very least a 40-Gallon Breeder Tank ASAP, as it's cruel to keep him in anything smaller, and you're going to continually have temperature gradient issues.
You can usually find very cheap 40 Gallon Breeder Tanks on Craigslist at any given time, and they typically come with a mesh lid. Otherwise, you can buy an Aqueon brand 40-Gallon Breeder Tank at any Petco for $50 that does not have any type of lid, but the lid isn't that important if you have an appropriate long UVB tube, as you can place the tube fixture across the top of the Hot Side of the tank, and then put the bright-white colored Basking Bulb in a clamp-lamp over the Hot Side of the tank.
In the meantime, you need to go and buy a lower-wattage, bright white colored Basking Bulb or a lower wattage, Halogen Indoor Flood Bulb immediately, as he's literally being cooked inside that tank right now. If you have a 100 watt basking bulb right now, I'd go to a 75 watt, if you have a 75 watt then I'd go to a 50 watt, etc.
