OP, we had very similar problems a couple of years ago. You've had some great advice here and I hope it gets sorted.
In case our experience is useful - we did claim liquidated damages for over-running and I was very clear that we would continue to do so, all variations were in writing, and the architect would come and do a valuation and issue a valuation for work carried out (ie each bit of the contract was assessed, he said 32% for plumbing or whatever, add it all up, and we'd pay up). The architect would definitely not let us pay for something that hadn't been done. He said some legit builders do need money upfront for supplies as cashflow can be an issue but if that is the case (it wasn't for us) it is best to place the order yourself and pay the supplier directly.
The builders did sort of finish but there were a couple of times when our architect warned us that pushing them could make them down tools, starting again with someone else would be messy and expensive and we might never get straight money-wise with the original lot. Sometimes they only had one trade on site, and things seemed to have ground to a halt, but then when that was done (eg the electrics) suddenly it was full steam ahead - for a bit. Again, the architect as very useful on that because he could say yes, that's fair enough, or nonsense, I'll send a warning email.
Our architect said our builders were not bad. Urgh. As well as the liquidated damages, they lost their contingency as they never finished/supplied documentation, but I don't think it mattered to them if by messing us around a bit they can take on a bigger job at the same time.
I didn't mean to write an essay but maybe another experience is useful!