Is there ever a state of thermal equilibrium between land and sea? What is the condition of the Land/Sea breeze at that time?
Thermal equilibrium suggests equal temperatures... in our active seabreeze season in Florida [early June - late September], that would mean temperatures generally in the low 80s °F (upper 20s °C)... which would be most common after sunset \ midmorning.
Ocean currents can mean some areas have much cooler (or even warmer) water than the land airmass during long periods of the year. So there are many places that have long stretches of the year when the seabreeze tends to stay quite strong (for example, the Pacific coast of California in the spring and summer) (it may be obvious for people in most areas, but I have remind myself that a seabreeze doesn't equate to thunderstorms in most places most of the time!)
Would expect that the conditions at such a time would reflect the synoptic conditions at the time (the larger-scale regional conditions caused by the usual interplay between transient high and low pressures, airmass movement, vertical instability and mixing, etc).
This doesn't mean the wind will definitely go calm when the sea and land are at equal temperatures... there's typically some regional pressure gradient [minimal gradients tend to be limited to broad high pressures areas and near cols]... and other factors, too; for my Florida situation, by midmorning winds often start to increase because heating is causing vertical mixing which brings stronger winds speeds down towards the surface from aloft. All that, plus the reduced friction over water, mean seashores\large lakeshores tend to be some of the rarer land locations to see calm winds, as this hourly August wind climatology of a coastal Florida site shows (they only see calm winds 4.5% of the time)
Likewise the wind can go calm at the seashore at times even when a large thermal imbalance exists... if the synoptic conditions oppose the local thermally-induced sea-land pressure gradient. A rough idea of the conditions that do this would be:
In the Northern Hemisphere, the synoptic winds should oppose the seabreeze if synoptic low pressure is off to the left \ synoptic high pressure is off to the right when looking at the ocean along a regionally representative shoreline during the daytime...
And the synoptic winds will oppose the landbreeze if synoptic low pressure is off to the right \ synoptic high pressure is off to the left when looking at the ocean along a representative shoreline during the night\early morning.
(Flip directions for SH)
Also it's worth noting that seabreeze boundary "fronts" can penetrate quite far inland and take time to fade away... even after temperatures have equalized at the shoreline or the landbreeze has started to develop near the coast... and if there are thunderstorms along the seabreeze boundaries, the cool air released by the rainfall often further reinforces the thermal gradient... I've certainly seen seabreeze boundaries persist deep into the night hundreds of miles from the coast.