The happy case is you have an instance of a linear type, you call some unrelated function which doesn't use that instance, then lexically later in the caller you do something with it. All typechecks easily, all is well.

If that unrelated function wants to throw an exception, reinstate some other continuation, or in any other way not execute the use of the instance after said function call, you have a design puzzle.

Affine types solve it by saying the object didn't need to be used anyway, so whatever. No type problems. Whatever deals with garbage collection will handle it.

Linear type systems usually solve it by refusing to let the called function branch to somewhere other than the nominal return site, aka no call/cc or exceptions in the language. Possibly function colouring to say no call/cc or exceptions in anything branched to within the lifetime of a linear variable.

The obvious answer to a compiler dev is that call/return is sugar anyway - all the variables "live across a call site" are implicitly arguments to the branch, so they're available tn whatever continuation the callee invokes. The plumbing is all fine. Typechecking is abstract interpretation so that's all good too, one can totally build the thing.

The missing piece is how you get the programmer to describe what they intend to happen to the linear instance on any continuations which are not the one denoted by the code immediately lexically after the function call 'instruction'. Where the syntax in question is an exception, it means "branch to somewhere implicitly recorded in a side table", so for the linear type use to work out you need to somehow annotate what is supposed to happen to the thing in that case. I don't know of a good notation for that.

(functions taking sum types are prone to being implemented as a branch on which field as active so you're definitely right that success+failure continuations can be transformed into one taking a sum - it's a reversible transform - but I don't think it helps the notation challenge).