Find a 2x4 and put another one or couple of additional screws in with plugs. Can’t seem to find any in your photo which could indicate that you didn’t put in enough screws. Each baseboard section should have at least 3 screws in our experience but you can have more if the wall requires it. 👍
Would you have a photo to share so we can take a look? Generally speaking if this is what you are thinking about we would not recommend caulking any gap. I guess its possible but then you have to deal with the caulk when disassembling the system later after you get water in the area. If your wall is not straight and there is a gap because of this then you add a few more screws in the baseboard to pull it in tighter if that makes sense.
The best solution is to cut the vertical strip to fit flush ... try not to run it behind the baseboard
A few gaps like this
Would you have a photo to share so we can take a look? Generally speaking if this is what you are thinking about we would not recommend caulking any gap. I guess its possible but then you have to deal with the caulk when disassembling the system later after you get water in the area. If your wall is not straight and there is a gap because of this then you add a few more screws in the baseboard to pull it in tighter if that makes sense.
Hope this helps!