Hey @sounak, sorry for the late reply.
There's no delay, however, the way we calculate the Insufficient funds is not the same as a getBalance. Actually, we sum gasLimit * gasPrice of every pending transaction, so we reserve the potential balance that might be used in the worst-case scenario.
That's why you may have Insufficient Funds errors while still having some balance in your relayer, it's just insufficient based on what's reserved for future transactions (although these transactions may consume less ether at the very end).
The best way to mitigate is to select an accurate gasLimit. I see you're using only 17% of your gasLimit (eg. this transaction)
Regarding the refill, let me circle it with the team since our funding address went out of funds. I'll get back to you once I have an answer 
Edit: I just talked with the team, and given rinkeby has been deprecated and Goerli funds are highly demanded, we decided to stop the autofunding until we find a reliable source of Goerli ETH.
You can get more Goerli ETH using these resources.
Best