What happens if full nodes don't check for double spends or deliberately lets a double spend TX pass and then this full node relays the transaction?
Since the miners trust these nodes to verify transactions, ... since miners don't check for double spends.


What makes you think this? Miners are also nodes; they are a subset of nodes. Miners still receive and verify all transactions, they don't just blindly trust what some other node has given them. Miners are incentivized to check transactions as including any invalid transactions means that the block will also be invalid and thus be rejected by the other nodes and miners on the network.

If a miner produces an invalid transaction (e.g. by including a double spend), the block will be considered invalid. Miners do not determine what is valid and invalid, nodes do. So when the nodes on the network receive that invalid block, they will reject it as invalid and not add it to their blockchain. Furthermore, since miners are also nodes, other miners will reject the block and thus they will not mine on top of it.

If miners mine on top of the invalid block, there will be a blockchain fork. The nodes on the network will refuse to follow a chain that contains an invalid block, even if it has more work. They only follow the valid chain with the most work, and an invalid block makes that chain invalid.