Maybe it is time to get a logic analyzer.
If you have to pick just one for now, IMHO a scope is more important, and more versatile, than a logic analyzer. But it's also more expensive to buy an entry-level scope than an entry-level logic analyzer. Only YOU can decide how to balance your budget.
Over the years I've developed and debugged countless serial interface systems with traditional, non-storage, non-decoding
analog {gasp} scopes. Serial comm has been around a lot longer than modern DSO's. Just work up in steps, by first repeatedly transmitting the same data packet - even a single byte - so the scope's triggering can give you a stable display. Confirm the bits are proper, then vary the bits to confirm the output tracks your code. Then get the receive side working. Now you're exchanging data at the byte level, so move to multiple bytes. And so on, up the curve. Soon you'll have serial data flowing around without a logic analyzer or a storage scope in sight.
Do fancy tools make it easier? You bet. Can it be done without them? Absolutely - many have done so before you, and you can do it too.