Lewis Burdett said he was trying to ‘carry on and be strong’ but wished ‘the nightmare would end’.
Ms Wellgreen, 47, was last seen in New Ash Green, Kent, on October 9.
She has not been in touch with her family since, and police say there have been no new transactions on her bank account or credit cards.
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Friends said she splits her time between Kent and Portsmouth, and that she works part-time at Puresun in Elm Grove, Southsea.
In an emotional post on Facebook, Lewis said: ‘I wanted to write to you today, it’s my birthday...but you know that. I’m trying to carry on and be strong but days like this really hit me. I just wish you were here.’
He added: ‘I keep looking at my phone waiting for you to ring or message even though I know you can’t and you cant be with me.
‘I’m finding it hard not being able to talk to you, to tell you our news, what we’re all doing. I even pick my phone up without thinking to ask you something.
‘I know I was lucky having you for so many years but I just didn’t think it would end so soon.
‘So when it’s quiet I just sit and have a cup of tea and a chat, I know you can hear me.
‘I will love you forever mum.
‘I’m going to share this letter with all the people helping to find you. So many people care about you and your in their thoughts and that helps me know we’re not alone.
‘I just wish this nightmare would end. Love Lewis.’
Family members have appealed for help to find Sarah on social media, while police have asked anyone with CCTV or dash cam footage from the day she disappeared to come forward.
A crowdfunding page set up to help Sarah’s family has raised £2,550 so far.
Sarah’s former partner Ben Lacomba, 38, from New Ash Green in Kent, has been charged with her murder and will go on trial later this year after pleading not guilty at a previous hearing.