Chester have also made their first summer signing, Solihull Moors striker Harry White, 22, who scored 12 goals in 2016-17, scoring in both of the Midlanders' two wins over City.
Shaw, 30, will combine his playing duties with helping boss Jon McCarthy.
Chester have also made an offer to coach Chris Iwelumo to remain.
The club are hopeful that the 38-year-old much-travelled Scot will continue to combine his coaching role with his media work.
On Monday, Chester announced six players would leave but offered deals to Sam Hughes and James Alabi.
The club lost their last six league games of the season to finish two points outside the relegation zone.

Thomas Sellar stabbed 23-year-old Jamie Walsh during a row in Greenock, Inverclyde, on 10 September 2015.
Sellar said he feared for his life but a jury convicted him of murder after hearing that the victim was unarmed.
At the High Court in Edinburgh, 38-year-old Sellar was ordered to serve a minimum of 18 years in prison before he can apply for parole.
Jailing him, judge Lord Ericht said: "It is because of your actions that Jamie Walsh is dead and it is because of your actions that the family of Jamie Walsh have lost a much loved son and brother."
During Sellar's trial, the court heard that he and Mr Walsh both lived at an address in Wren Road, Greenock.
Mr Walsh had recently moved into a flat below Sellar, which was being rented by the victim's friend.
In the days leading up to the murder, Sellar had objected to "noise" which was coming from the downstairs flat.
On the night of the incident, Sellar said he had gone to the downstairs flat to try to convince the residents to keep the noise down.
He claimed that a group of youths objected to this and had "battered" him and chased him back to his flat.
Sellar claimed he felt he had no other option but to grab a knife in a bid to protect himself.
He said that in the ensuing fight, the knife came into contact with Mr Walsh, who died shortly afterwards.
He said: "They chased me down. I couldn't breathe. I never intended to kill him. I never intended for anything like this to happen."
Christopher Walsh told jurors that his brother did nothing wrong and that Sellar had attacked him.
He told the court that following the incident, Sellar claimed that Mr Walsh had stabbed him first and showed off a wound which he claimed he had suffered during the fight.
He said his brother did not have a knife and when asked why Sellar claimed to have been stabbed by his brother, answered: "He was trying to get a defence."
Sellar denied stabbing himself and maintained he was acting in self defence.
However, a jury refused to believe his claims and returned a guilty verdict to a murder charge.
Following the verdict, it emerged that Sellar, had six previous convictions for offences involving knives.
He also wrote a two-act play - Dead Boys tales - when he was 19-years-old.
The work, which was performed in Greenock in 1997 by a cast of young people struggling with addiction - depicted the problems teenagers had in staying away from drugs and crime.
Chester midfielder Tom Shaw has been appointed player-assistant manager at the National League side after signing a new contract at Bumpers Lane.
[NEXT_CONCEPT]
A man who claimed he killed his neighbour in an act of self defence has been jailed for life for murder.