The Reds are top of Welsh League Division One, two points ahead of County on level games.
Briton Ferry Llansawel v Cambrian and Clydach plus Cwmbran Celtic v Goytre United have also been postponed because of the cup ties, while three games are due to go ahead.
They are Monmouth Town v Port Talbot Town, Undy Athletic v Afan Lido and Goytre FC v Ton Pentre.
Llanelli chairman David Craddock is going to step down from his position at the end of this season.
He took the role in 2013 when Llanelli reformed after liquidation.
Craddock, known as DC, took up the reigns and stabilised the club using his business prowess, an endless list of contacts and his drive to lead Llanelli Town AFC to the highest place possible in Welsh football.
Club official Bob Jeffries: “To say it has been an eventful five seasons is an understatement.
“When I first approached DC to take over the Reds we did not even know what league we would be playing in. It was always thought the we would be in the Carmarthenshire league and staff who stayed devised a five year plan to get the Reds back to Welsh League Division One.
“When DC came on board the first thing we did was pick a name for the new club and Llanelli Town FC was formed.”
The Reds were eventually placed in Welsh League Division Three, they have climbed back to the First Division are are currently top.
“The towns football community will forever be indebted to David for not only keeping the football team going, but for making dreams become reality over the last few years, with potential promotion to the WPL so close,” says Jefferies.
“Thanks to David, the club has and will continue to be in the black, with no outstanding debts. He has always insisted that the towns football team should be for local talent and this is a philosophy which has been instilled within the academy.”
Investment is needed at the club and and Llanelli can be contacted via Wayne Stephens on his e-mail Director@llanelli.academy.
Welsh League Division One fixtures today: Briton Ferry Llansawel v Cambrian and Clydach, postponed; Cwmbran Celtic v Caerau, postponed; Cwmamman United v Goytre United (2pm), Goytre FC v Ton Pentre, 2pm; Llanelli Town v Haverfordwest County, postponed; Monmouth Town v Port Talbot Town (2pm); Undy Athletic v Afan Lido (2pm).