Four out of the seven members of Cosla’s new political team are SNP councillors, including the high-profile resources brief.
Cosla has been highly critical of the Scottish Government’s spending plans, accusing it of failing to respect the role of local authorities.
It warned last week: "As things stand, the only option available to councils is yet fewer jobs and cuts to services that are essential to communities everywhere.”
Scottish Conservative local government spokesman Miles Briggs said: “This news is another hammer blow to Scotland’s savagely underfunded councils.
“Until now, Cosla has been rightly vocal in calling out the sustained and systematic budget cuts local authorities have been handed year after year by the SNP Government.
“It’s hard to see that continuing with an SNP councillor as president.
"I fear Shona Morrison will prove to be just the sort of lackey Nicola Sturgeon wants in charge – someone who will meekly accept unfair funding deals and not dare to criticise her Government for imposing them.
“Scotland’s councils are on their knees, barely able to provide lifeline services to local residents.
"They need a strong champion to demand the kind of fair funding mechanism that the Scottish Conservatives would enshrine in law.
"Instead, they now have an SNP cheerleader at the helm.
“It is abundantly clear that local government in Scotland faces an SNP-Green coalition intent on centralising powers and cutting funding from local councils.
"Scottish Conservatives will work to defend our communities and fight for a better deal.”
Scottish Liberal Democrat local government spokesman Peter Barrett said: "The risk with the SNP taking on these key roles is that so often their representatives are simply apologists for SNP Government cuts to local authority budgets. It's like putting the fox in charge of the hen house.”
Steven Heddle, an independent councillor in Orkney, was appointed Cosla’s new vice-president.
Ms Morrison said: “I am honoured to be elected president of Cosla and excited to take on this opportunity in one of the top political jobs in Scotland.
“I look forward to continuing the progress which has already been made on a cross-party basis over the last five-year term.
“I am clear about the pressures faced by local government and the challenges and opportunities ahead.
"This is an important job and a challenge I will relish and give my all to.
"I am a politician and a member of a political party. However, my focus in this role will be on the ‘team local government’.
“Working with the vice-president and the spokespeople at Cosla, I’m confident we will deliver on our key priorities as we work together to be the voice for local councils and champion the essential services they continue to provide under increasingly challenging circumstances.”
Mr Heddle said: “I am delighted to be elected to the position of Cosla vice-president.
"It is a wonderful opportunity and a role that I am honoured to take on, on behalf of the whole of local government.”