Connecting to random peers for privacy is probably not a good idea. What you should do is what Tor does: pick random peers once, and use them as long as possible. Tor calls these your guard nodes.

To understand why, consider what happens if you do n transactions. Each time you do a transaction, if you pick a random node each time, there's a chance that the node you pick is evil. These random chances add up, and eventually, you've probably sent a transaction through an evil monitoring node. Not good.

Meanwhile, if you use as few distinct nodes as possible, since (we assume) the vast majority of nodes are good, and not logging, you're unlikely to ever run into an evil logging node.

Of course, you might get unlucky too. But the bad guys determining your identity once is probably enough.