The 31-year-old scored just twice in 36 appearances in the Dons' maiden League One campaign last season.
He now drops down to League Two, where he helped Shrewsbury to promotion in 2014-15 and finished top scorer at both Crawley Town and Macclesfield.
Barnett, now on the 14th club of his career, also played in the Championship for Peterborough and Ipswich.
He made his name in non-league football with Rushall Olympic and Hednesford Town, after being released as a youngster by West Bromwich Albion.
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Stephen Mallan converted a penalty to give the Buddies the lead after Darren Cole had fouled Calum Gallagher.
Liam Buchanan quickly levelled for Livi but Shankland's strike from outside the box had the visitors ahead again by half-time.
Jordan White poked home for the hosts only for Shankland to fire an impressive volley and settle the match.

Stocktrade is being transferred to the Dundee-headquartered firm for £14m in cash.
Its most recent annual revenue was £9m, with £1.3m in pre-tax profit.
Last September, Stocktrade had £4.3bn of assets under management. The buyer said it now stands at £4.6bn, bringing 48,000 new customers.
That should increase the scale of Alliance Trust Savings to £11.5bn in assets under management, with 105,000 customers, allowing for more efficient operations.
It will also increase the Dundonians range of services, including telephone and online share dealing services for pension providers, building societies, company share schemes and corporate Individual Savings Accounts.
Brewin Dolphin said the sale would only clear a net gain of £1m, after paying for separation costs.
The deal still requires clearance from regulators, and is expected to close during this year.
Alliance Trust Savings was part of the strategy last month criticised by Elliott Advisers, the hedge fund and activist investor in parent company Alliance Trust.
In criticising the Dundee company's management, it succeeded in getting two new directors appointed. They could have sought to block further expansion of the financial retail platform. But it appears, from this deal, that they have agreed to expenditure on growth.
Alliance Trust believes that the expansion could help its savings division into profit by 2016, apart from the cost of integration.
Patrick Mill, managing director of Alliance Trust Savings (ATS), said: "Stocktrade is a great strategic fit for ATS, giving us access to new customers, an increase in assets and the opportunity to build a strategic partnership capability.
"It also gives us the ability to align a traditional stockbroking business with our innovative, scalable and cost effective approach by offering customers online access alongside our in-house customer service team."
Port Vale have completed the signing of striker Tyrone Barnett from AFC Wimbledon on a two-year deal.
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Lawrence Shankland scored twice as St Mirren beat second-bottom Livingston in the Scottish Championship.
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Alliance Trust Savings has announced plans to buy an Edinburgh-based share trading company from Brewin Dolphin.