mom-6

Depending on the actual style of t-shirt that has the cap sleeves you might possibly get some of the really wide embroidered eyelet "lace" and create a longer sleeve. I've seen some that would look wonderful on and others that it would just look silly, so I don't know if this would work or not.

For the button down shirt you would need to find a coordinating fabric to add a strip to the sleeve either at the inside seam or down the middle of the sleeve. If you do the middle of the sleeve you could even put in a strip of lace. Either way would require taking the sleeve out and redoing it. (And why do some shirt designers figure we don't have muscles in our arms just because we're female?)
I understand your dilemma as I usually have to get guys shirts for the sleeves to fit right. I hate ladies long sleeve shirts because they are both too skinny and way too short. The long sleeve T-shirts end up as three quarter length on me. So the wrist hits at a much wider part of my arm. Yuk!

Greenheron

When my husband 'out grew' one of his favorite shirts with an embroidered logo, I cut out the logo and appliqued it to a new, larger shirt. Did not look as good as the original but he liked it.

diamondee

Too bad you couldn't take the logo off the shirts and applique it to a shirt that fits right.

looks like greenheron and I had the same idea

Pepita

My husband has a fishing or hunting shirt with 'gussets' in it. These gussets are in places like under the sleeves, the yoke of the shirt and I think down the seam line. The gussets have a T shirt type of fabric that have the little holes in it, so that the shirt is cool. I think if you can see one you might be able to re-fashion your shirt so that it is more serviceable to you.

IBQUILTIN

It would be simple to get some contrasting fabric and replace with a dolman sleeve