A # 1 spruce 12x8 joist (@ 16" o.c.) will support only up to a 12'-6" span for living areas of up to 40 lbs per sq ft.
These existing joists you have (even tho they are a full 2x8 in size) are not strong enough to support this 16' span, and that is why they have sagged. Rather than putting in a supporting beam and posts why not just "sister " each joists with a normal 2x8. it would probably be just as costly but it would still keep the area "open" without any beam/posts in the way.
When installing each "sister" joist make sure you use a temporary support to take some of the sag out of the existing joists before securing the "sister" joist to the existing ones. Do one joist at a time.