All three players are contracted to the League Two Sulphurites until the summer of 2023 but were this week told that they are free to move on.
Weaver has also opted against offering new terms to eight other first-teamers whose deals are due to expire imminently, the first step in what looks set to be a huge overhaul of his squad following an underwhelming 2021/22 campaign.
Thus, with somewhere in the region of a dozen players potentially due to arrive at Wetherby Road over the coming months, the Town boss said he felt it was only fair to be "up front" with those members of his existing group whom he anticipates will find themselves on the fringes of things next term.
"I initiated the conversation, so I have to take responsibility for the decision to put them on the transfer list, but I think the players understood the thinking," Weaver told the Harrogate Advertiser.
"From our point of view, we are trying to visualise how we improve this squad. One of the big things last season was that we conceded too many goals, so defence is an area in which we have to get much better.
"We are trying to be decisive in our planning, but also trying to be fair to these lads. We've been really up front from the earliest possible moment and told them that we don't expect them to be playing regularly here next season.
"Telling them that now, rather than keeping them hanging on gives them the best possible chance of finding a new home and going somewhere they have a better chance of starting games week in, week out.
"They've got all summer to work something out now, and we will help them with that as much as we can. If we had waited until we declared our 22-man EFL squad in August and these lads found out that way that they weren't in our plans, then they'd have limited time and options. I don't think that would have been the right thing to do."
The decision to transfer-list veteran centre-half Legge and powerhouse striker Martin comes as no real shock.
Legge, 36, only arrived at the EnviroVent Stadium from Port Vale in January, but after featuring in seven of the first nine games he was available for, he made just one substitute appearance after February 22.
Martin, 30, was loaned out to National League Halifax before being recalled due to Town's late-season injury crisis and has fallen well down the pecking order having played 42 times last term.
Opting to give 24-year-old former Liverpool trainee Sheron permission to move on was more of a surprise given that he started 15 of Harrogate's final 16 matches of the campaign and the fact that he can operate in four different positions.
"I felt particularly sad breaking the news to Nathan Sheron because he came here and signed a two-year deal and he's had some good games towards the back end of the season," Weaver admitted.
"But, we need to strengthen the defence and I'm not quite thinking of him as a player who will get regular minutes next season.
"He is 24 and at that age, staying here and hardly playing next season wouldn't be any good for him in terms of his development.
"He's got three months now to find a new home, and I'm certain there will be plenty of interest because he's a player of good pedigree, he's so versatile and such a great lad."
Goalkeeper Joe Cracknell, defenders Ryan Fallowfield and Lewis Page, midfielders Simon Power, Connor Kirby and Joe Leesley and striker Mark Beck were the Town players not to be offered new deals at Wetherby Road, while the long-serving Lloyd Kerry had already announced his intention to retire from playing.
Veteran centre-half Rory McArdle is also out of contract, but is currently negotiating with the club over extending his stay in North Yorkshire.