What you could try is modifying the custom shape generator codes to increase resolution. What it boils down to is finding the parameter or other equation statements that define segmentation. For example, in the custom cylinder code you can find this:
var ndivs = Tess.circleDivisions(Math.max(r1,r2));
which can be changed to something like
var ndivs = 2*Tess.circleDivisions(Math.max(r1,r2));
Don't go too crazy with the resolution improvements though. Tinkercad will choke on too much resolution improvement.. That can manifest itself in slow response times and more grouping errors. Larger resulting file model downloads can become a problem as can transfers to Shapeways. Exports can freeze up, time out, or mysteriously send multiple copies to your Shapeways account. And as pointed out before printer resolutions limit the improvements you would realize so there's no need for excess smoothness.
If you already have designs using the standard un-editable elements you would need to dis-assemble your designs and swap in corresponding editable shape generator elements for the critical elements. You can mix and match the older and custom elements. This is a bottom up type of change. If you want to smooth models after they come out of Tinkercad that would need to be done in a different editor. I have no good experience in that realm.
I don't know all the ins and outs of the custom shapes. I don't know if they end up being global elements or if they become unique within a saved model. If they don't have a basic instruction document there are some posts in their blog about the feature. You can also generate your own custom element codes if you know what you're doing and share them with the Tinkercad community.