Those certainly are the key indicators of a leaking heater core.
See "Related Questions" below - not only the one for the 1986-1995 Taurus/Sables, but read through the 1996-1999 heater core question - some of the information is relevant to the earlier generation too.
Note: I've done this - my 1987 wagon - tough job, but certainly doable. Be sure to do your stretching exercises before starting - lots of contortions and bending.
Check the coolant level. My 2000 Ford Taurus has a constant flow cooling system and the heater will not work if the coolant level is too low.
probably the heater core is clogged
The Taurus does not have a heater control valve. Coolant flows through the core at all times. Be sure to review the "Related Question" below for much more about Taurus temperature controls
Low coolant? Thermostat not opening?
Check the coolant in the 96 Taurus. Check to insure that the thermostat is working properly, and that the water pump is working.
The heater core could be clogged up.
Do you mean that your heater core is leaking coolant ? That would mean that the heater core is clogged. The heater core is like a mini-rad, if it is clogged it leaks.AnswerNo, there is not an overflow tube/outlet on the heater core.See "Related Questions" below for more about Taurus / Sable heating/cooling systems.
the heater core could be pluged or the heater control valve is stuck closed
heater core may be partly plugged .
either low coolant or a thermostat stuck openSee "Related Questions" below for much more trouble-shooting and diagnostics for Taurus/Sable "no heat" problems...
The recommended coolant type for a Ford Taurus is the green coolant. There are several different brands such as Prestone.
Could this be 2 separate problems?Stuck in defrost modeNo heatSee "Related Questions" below for more about both