My bathroom sink has been slow going down so we put some drain cleaner yesterday which we've done before with no issue. All of a sudden today I hear water leaking downstairs so I used a little hand plunger to try and clear it and a bunch of black luqid came up but it's still leaking.

What else can I do about this? It's like coming from the floor downstairs so I'm assuming it's the pipes behind the wall that are leaking. We definitely don't have the money to have it all redone so we need suggestions of what we can try that could possibly help? Would snaking it cause further damage?

Side note, the house was built in the 40s and I highly doubt anything has been changed since so whenever the toilet gets clogged, water leaks from the pipes downstairs but once it's plunged and cleared the leaking stops. That's why I tried plunging the sink.