Last time I took it out, it had severe trouble stopping.
One minor clarification first. One thing that many people don’t appreciate is that our brake and shift cables require periodic replacement. Dirt eventually intrudes into the housing. It is not a scam, it is an inherent property of the system. Is there an alternative? Yes, electronic shifting, where you put motors on the derailers and you actuate them with wired signals or wirelessly. That stuff works very nicely, but it is very expensive, and it is less likely to break but if it does you sometimes have no idea what went wrong and it can’t be fixed in the field.
Basically, if you changed the cables, you might be surprised. You can also benefit from upgrading the stock brake pads, sometimes. Kool Stop is often recommended. It is true that disc brakes have more stopping power than rim brakes, especially in the wet. However, most of us don’t need the extra stopping power, and most of us don’t ride in the wet that much, and good (and well maintained) rim brakes can stop well enough in the wet.
Moving on, in principle, you can move all your current 105 parts to the new bike save the brakes. Many people tend to recommend that if you’re going disc brakes, they might as well be hydraulic. I tend to agree, but cable-actuated hydraulic disc brakes exist, and my understanding is that they are quite good. Whatever the case, if you are going with cable disc brakes, you would want good compressionless brake housing for them. If hydraulic brakes, you’d need new hydraulic levers. At the 105 level, those would inherently be 11 speed, so you’d need at the very minimum a new rear derailer, chain, and cassette. The front derailer should work acceptably, and the crankset is very likely to work even if it doesn’t shift quite as well as a fully stock 11s setup. For the front derailer, there may be a complication. FDs come for braze on mounts, or they come in 3 clamp sizes (31.8, 28.6, and 34.9). I have a feeling your Specialized is 31.8 and the Surly may be 28.6. You’d need to double check this. if braze on, you can get a clamp adapter for any of the 3 sizes, and mount the braze on FD to the clamp. This only works with round seat tubes, but I assume that both bikes are round.
Speaking of chain and cassette, by now, I would expect yours to be worn out, so you might want new ones anyway.
You won’t be able to move your wheels over, as they don’t have disc rotor mounts. They may be too lightly built for a bike loaded with cargo anyway.
The handlebars, stem, and seatpost should be able to fit. However, for the stem, the Surly is likely to have a different reach and stack than the Specialized, and you’re likely to want a different length stem.