I am worried about your solicitor. I don't know the answer to this question, but I would have thought that a solicitor should be able to advise without consulting a barrister unless there is something very odd in this which you have not told us. However, if the solicitor has decided that there is nothing you can do, then he might as well run it by a barrister for a second opinion.
Neither the solicitor nor the auctioneer is a party to the contract, so they would not sign.
The normal process is
The vendor's solicitors send you the contract.
You sign it and return it to them
They sign it and return a copy to you.
Your solicitor should have noted that the contract was not signed by both owners, but if he had noticed, what could he have done? Simply sent the contracts back to be signed, so you have not lost anything by their oversight.