Although there are similarities, the landlord is the same as an ordinary debt collector. If the collector comes knocking and you present to him what he claims you owe him in cash or in other good funds, the collector would be foolish to refuse to take it. If he refused, then you should be able to convince a court to avoid further award of interest and penalties. On the other hand, if you gave him an personal or business check for a large amount and the collector refused to take it, that may be another issue, especially if you have a history of bouncing checks.
But the landlord is in a different situation. If you have been in default of your rent, he has the right to evict you, even if you later come up with every penny of the back rent and other charges. Eviction does not wipe away the debt of the rent owed. Sometimes leases include grace and cure periods, and in residential leases state law may provide protection against a landlord seeking to use a single late rent payment as a pretext to evict.
So suppose you come to the landlord and say here is everything I owe you, will you allow me X number of days to remove my property? The landlord would be within his rights to demand the amount for those days you need. But if you have a history of not being able to keep to your promises about moving out, he may refuse.
So the problem becomes his premises, your property. One remedy would be to bring a tort action, suing for the value of any property converted by the landlord over and above what you owe and seek the equitable remedy of replevin to have the court order the property restored to you. If the property is unique and irreplaceable, you may be well-advised to seek a temporary restraining order/preliminary injunction against the landlord to stop any further sales. This may not be possible if there is an outstanding amount and the landlord is selling pursuant to a judgment and execution.