Disclaimer: I do not practice Aikido. I practice Tai Chi. However, from what I have seen of the two arts, they have enough similarity that I feel bold enough to try to suggest how Aikido would handle a situation. My goal is to be helpful to someone whose problems sound very similar to the hard parts of learning Tai Chi.
In addition to mattm's answer of "sung," which I agree with, another thing to consider is your frame of mind. What I have found is that one tenses up when one feels the need to force their will upon another. Tense muscles are in a better position to apply a great deal of force along a path which you predicted ahead of time.
I do not do Aikido, but if you are receiving helpful advice from an expert to relax your shoulders, I think I can read between the lines and recommend. You are probably trying to throw your opponent. The simple version of how to fix that is to instead let your opponent throw themselves, all you do is guide the way. The advanced version will involve varying phrases of merged qi, where instead of it being you throwing your opponent or your opponent throwing themselves, it becomes "you and your opponent throw him." However, that takes some time to get to, and I would let your instructor guide your imagery and terminology.
As for avoiding the strength fight, that becomes possible as you start talking about qi. I give that disclaimer because many do not believe qi exists, and are quick to help others believe it does not exist as well. However, discussing Aikido without qi is an exercise in futility, because the art is built around such terms.
The key to avoiding those strength matches is two fold. First step you have covered: don't want to get into strength matches. The second step is the obviously tricky one: prevent them from forcing you to enter a strength match. What Aikido will teach you to do is "meld your energy" or "meld your qi" with your opponent to redirect their energy around you. Then they will not be able to take control of your body with those strength moves.
I have to mention the qi wording because that is the one Aikido will teach, and I don't want to direct you in a way that leads you off of the path of your art. However, there are well known arguments for why you shouldn't use qi based answers on Martial Arts SE, so I can try to give you some English explanations of what you are looking for. Hopefully they can guide you until the wordings from your instructor take you the rest of the way.
It is reasonable to divide a grappling situation like your bear-hug strength fight into 3 period. The first period is all visual. You may see him running at you, but no contact has occurred. This changes at first contact. Once there is a point of contact, he knows where one point of your body is, tactility. This is an important transition. Visual processing is slow. It can take 100-300ms for a change in visual information to properly register mid-fight. By evolutionary necessity, tactile information is processed much faster, particularly sensormotor interactions. Consider how fast you must respond when you put your hand on a hot stove. I can't find numbers off hand, but I remember seeing them in the 10-50ms range.
The third period is after the point of application is found. A point of contact lets them know where a point is. A point of application lets them know where limbs are and gives sufficient control to manipulate them. This is when the grapple is complete, and you are bear-hugged. Now he knows where your body is, and how you are trying to shift your weight, and he can adjust accordingly.
The middle period is key. You want to give them a point of contact, but never a point of application. Contact with an expert in aikido is a hidden doom: the information from that point of contact goes both ways. Aikido, as best as I can tell, is one of the arts which excels in learning more information about their opponent than they give to the opponent, by subtly adjusting their musculature to make it hard to feel what they are doing. They lead the opponent to believe they are going to complete the bear-hug if they overextend just a little bit more. Meanwhile, they are fully aware of exactly where their opponent's center of mass is, and exactly how balanced or unbalanced they are. Soon, the opponent's defense mechanisms have to kick into play, keeping the opponent upright as they lose their balance. If those mechanisms engage the opponent's arms, it becomes hard for them to bear hug.
Once the opponent is off balance and, for lack of a better word, diffused, the Aikido master is now the only one who really has control over any muscles. All of the opponent's muscles are busy trying to avoid falling. At this point, the Aikido master uses qi to send the opponent flying. Call it what you want, but I think the nature of what is being done can be approximated in English: The Aikido master basically offers the opponent's defense circuits help. After all, they're overworked. They happily treat the Aikido master as a resource to regain balance, but the master is smarter than that. He fanagles the defense mechanisms into believing they are in a situation they aren't, where the best answer is to do something like stiffen the legs. Now the opponent literally tries to jump over the master, not even realizing that's what he's trying to do. The Aikido master now has the easy job: he simply obliges, and lets the opponent fly.
I know there's more to it than that, but that first pass version may be enough to show you why you don't want to tense up (it reveals information), and how you will be trained to avoid the bear hug (you allow a point of contact, but never a point of application).
Of course, that is all assuming you want to cut it razor edge close. Otherwise, a kick to the balls is a pretty effective method of saying "no hugging." However, training to deal with that razor thin edge turns out to be a pretty effective method of improving one's control outside of a fight as well. The best argument I have heard for martial arts is not what it does for you when you get in a fight, its what it does for you when there is no fighting that needs to be done. Every martial art has that in it, but with arts like Aikido, I find it more evident than I do in other arts.