When fan switch is set to the "On" position fan comes on and system blows hot air. When switch is set to the "Auto" position fan does not come on despite temperature being set to call for heat. While fan is running with switch in the "On" position, if I slide switch to "Auto", fan shuts off. Is it time to replace T-stat?

Thermostat is probably fine. When the thermostat is set to auto for the blower it operates through a control in the furnace called a fan and limit control or fan control. This control allows the unit to heat up before the blower comes on and in your case it seems not to be working correctly. I would suggest checking and if defective replacing the control for the blower that operates in the heat mode.

NOTE when the fan is set to on it bypasses that control and a relay just brings the fan on. This is why your fan still works when you select ON instead of auto.