Yeah, it’s a bad habit that a lot of traditional arts teach and unfortunately one you’re going to have to unlearn. I will say though that dropping the non punching hand when you throw “power” strikes is something that is fairly common for people learning how to box; they tend to keep it up ok when throwing the jab, but tend to drop it on the rear hand or lead hook.
With the round kicks you will get numerous “correct”'ways that people throw them or teach them. Some people teach to “pendulum” the arm down/back to add extra power, others like to keep both hands “up” and will just pull the elbow slightly back/down (Kyokushin fighters often teach this), yet others like to extend the arm to obstruct incoming counters/blind distract opponent, or trap (my preferred method).
Regardless of what you do with the “lead” (arm on same side you are kicking with) arm there are a few things you can do to minimize your chances of being countered hard:
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angle out-if your head is between your opponent’s shoulders when you land the kick there is a possibility of eating a big counter (straight right if you are throwing a right round kick), so to avoid this you can step/angle slightly outside your opponent’s right foot or “turn in” so you are flanking your opponent to their left before throwing the kick
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Roll your lead shoulder (like doing s shoulder roll defense) and keep your rear hand (left hand if throwing the right round kick) up in front of your chin to catch anything that might slip by your shoulder. This also ensure full hip rotation on the kick
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set the kick up right-just throwing a lead rear leg round kick can work if you are fast and have conditioned your shins (or just naturally have hard, sharp, dense bones) to taking impacts on hard surfaces, but there is also a much greater chance of running into a shin/knee block, forearm/elbow, or having someone step in, check your kicks (which is not the same thing as a shin block although people often use the term to mean that) and trade your leg kick for a punch to your body or head. Again, not saying this can’t work as there are plenty of fighters who do it with success, but it’s riskier than using the kick as a “power” follow up shot (like you would use the straight right). A couple good combinations that set it up well and make it harder to block include:
-Jab, straight right, right leg round kick (neither the jab or straight right has to land necessarily, but with will serve to get the opponent’s attention “upstairs”, both can temporarily blind them from seeing your kick coming if you place your punches at mouth/chin/neck level depending on range, or 1 or both can be used to “trap” the opponent’s lead arm and bodyweight onto their lead leg (making it difficult to lift the leg to block your kick. The straight right also has a nice way of pre-turning your hips and flowing very easily into the right round kick making the combination quick, hard to spot, and good against people who like to shoulder roll or “run”.
-(jab, if using offensively, though not necessary) straight right, left hook, right round kick- this kick can be thrown more power style or more speed style depending on how you execute it. If throwing it power style then the end position of your left hook should put your weight on your right leg, thus “loading” that side to allow you to powerfully push off that leg to initiate a power right round kick (it is also advise able to take a slight step with your left leg “outside” their right shoulder/leg as you initiate the kick to provide better protection from their counter right hand and to add mass transfer to your kick. The other way to use this is to use the left hook to disrupt their balance (aim high on the head, it’s more of a “bump”/slap than a punch meant to hurt them) and put their weight on their left leg, then you throw a switch speed rear leg round kick (essentially you are throwing the kick at the same time you are doing a switch step) to their left leg. This won’t have the same thunderous impact or do the same amount of damage as the power version, but if done right it’s very safe and hard to block.
-lead drag step inside round kick (which when mastered is a great lead off technique and very hard to block, but requires some practice and athleticism and for a guy your size may not be the most practical lead off technique in which case you could use Grandmaster Bill Wallace’s “no step” footwork trick to set it up), repeated several times till your opponent starts really trying hard to get their leg up to check your kick as soon as they see your lead hip “cock”, then fake the lead kick (you can either do a “knee fake” where you actually throw a half kick/raise the leg off the floor or just a “hip fake” where you cock the hip like you are starting the kick but don’t actually throw the leg at all) to draw their reaction and immediately switch to a rear leg round kick aimed at their rear supporting leg (sometimes you go through their lead leg and kick both legs out, sometimes you go under their lead leg and just hit the rear one, depends on how high they lift their leg to block you and how low you aim your kick.
There are plenty of variations on these but these are some simple template combos that you can use on the bag and do directly translate to sparring/fighting.
Hope this helps.