If I understand well, your biggest issue is how to present processes in psychrometric chart.
And that is really the issue, you should know which process you apply, than you will know which line or curve to apply.
For sensible heating, it is rather simple, and that is horizontal line you mentioned.
For cooling first you need to know whether you have dry cooling or not. As you mention your desire to lower humidity, apparently you deal with wet colling, than process line is not straight, it is curve which depends on cooling coil construction, and you can get prcecise result from manufacturer only.
Commercial fan-coils and similar devices will not have anything avaialable, as they are not intended for regulated dehumidification, so manufacters usual give some rough data on moisture removal.
You could construct process curve from their data if they would give several process points, but normally they declare moisture reomoval at referent conditions only.
Try this, it gives nice overwiev of curves related to processes:
Pitty Trane and Carrier don't bother to provide such simple explanations while offering software for precise calculations.