Regardless of whether you believe Paul is gay or not (personally I believe he was bisexual) Paul loved Linda. They met for the first time in 1967 and then met up again in May 1968. Paul and Jane were no longer engaged so Paul might have been looking for another girl to keep his and Johns relationship a secret. However, rather than with Jane - he fell in love with Linda. Paul had wanted children (which he couldn't have with John) and Linda had a daughter Heather. He invited them to move in with him.
One time he was feeling guilty for being tired and she replied, "it's allowed," which amazed him. "I remember thinking, Fucking hell! That was a mind-blower. I'd never been with anyone who thought like that ... it was patently clear that it was allowed to be tired."
Perhaps his relationship with John was starting to become deeply one sided.
When the Beatles broke up, Paul became deeply depressed and Linda helped him through that.
Linda was probably Paul's second soulmate.