We say "mucho" to express an amount. On the contrary, we use "tanto" sometimes to exaggerate something and transmit that you've danced so much. It is always followed by suspension points and it is an unfinished sentence. It is correct to say "Estábamos muy cansados porque bailamos y cantamos mucho" because you express an amount of time. If you want to use "tanto" you should put suspension points or change the order: "estábamos muy cansados. Bailamos y cantamos tanto..." or "Bailamos y cantamos tanto que nos cansamos".

In general you use it like that: - Tenemos "mucho" tiempo para estudiar. (We have a lot of time to study.) But: - Tenemos "tanto" tiempo para estudiar "que" podemos descansar un momento. (We have such a long time to study that we can rest for a moment.) Basically "tanto" is usually followed by "que". The difference is that "mucho" just gives an idea of a large amount, while "tanto" gives an idea of a large amount, and also we have to add another action. If we say: - Tenemos "tanto" de qué hablar. (We have so much to talk about.) It means that we have a lot of things to talk about, and that maybe time won't be long enough, or that we should continue another day, etc. - Tenemos "tanto" de qué hablar "que" sería mejor organizarnos. (We have so much to talk about that we should organize ourselves.) - Hay "muchos" ejemplos para darte. (There are many/a lot of examples to give you.) - Hay "tantos" ejemplos para darte "que" el mío solo no alcanza. (There are so many examples to give you that mine alone isn't enough.) I hope this helped you :)