It is always helpful to describe the problem you are trying to solve rather that asking for a specific solution.
These two objects can be very different in their properties and how they can be used. Hopefully we create new objects only where there is a real need or purpose. We also have to consider how an object will be exported to our field devices and transfered to other CAD products. An alignment can have a station equation and spiral curves and circular vertical curves while a linestring cannot. Each has limits and intended uses.
You cannot offset a vertical curve along a horizontal curve and hold both the grade in or out and the cross slope of a cross section. As the length of the line changes with the offset in a HAL curve, something has to give. That is why the offset command creates straight lines along the parabolic curve. You can tighten up the number of points created with the project settings>computations>Surface>breakline approximation parameters. Change the horizontal and vertical tolerance to be 0.001 to get more points. You can use the geometryreport command to check the specific values.
Ideally you would leave things parametrically defined. You can use the alignment to create a surface and tune the surface to vary the density when needed. Your template can define the offset nodes and they will vary with the density settings.
Once you offset the alignment you now have dumb CAD data. It may have a purpose, but now you have to manage when it is applicable and if you change your design, you have to remember to create a new offset. If you really just want a copy, then use the copy command.
One small trick that might help. Offset the HAL. Edit the hal. Create a new VAL and use the append to alignment option to use the offset line as the basis for the VAL. Create a new profile view. If the new val is not at the right station, use the Move command to move it from its current beginning to the desired station and offset. The new VAL should be the same elevation as the old VAL, at each of the VAL points created. A HAL can have multiple VALs.
With a little more definition of what you want to do with the offset linestring, we might find you need something different than what I described above. Hopefully this helps.