Both are grammatically correct, and each communicates a subtle distinction. The word amount is used in the singular to convey a single quantity:
1.0 A quantity of something, especially the total of a thing or things in number, size, value, or extent:
sport gives an enormous amount of pleasure to many people
the substance is harmless if taken in small amounts
1.1 A sum of money:
they have spent a colossal amount rebuilding the stadium
ODO emphasis added
Since quantities tend to be measured with reference to content, source, time and space, a heterogenous amount is often presented as the plural amounts, to communicate the complexity of content, source, time or space. Consider the expression large amounts of data, where various types of data comes from various sources at various times, and are stored in various places, but they are considered together:
In other words, they are two datatypes designed to hold really
large amounts of data. Blobs, represented by the BLOB datatype, hold large amounts of binary data. Similarly, clobs, represented
by the CLOB datatype, hold large amounts of text data..
Database Programming with JDBC and Java by George Reese, emphasis added
All of these large amounts of data could have been conglomerated into a single large amount of data, but the pluralization communicates a sense of the complexity within that single large amount.
The same perception of complexity is true with large amounts of money:
One way and another, large amounts of money leave the West
household on an almost daily basis, but it was ever thus and Richard
keeps fairly close track of each individual's expenditure.
Relationship Fundraising by Ken Burnett, emphasis added
The large amount of each individual's expenditure could be tallied together as a single large amount of money, but the pluralization expresses a sense of the complexity. Even the large amount of an individual's expenditure can be expressed as a plural to communicate the various budget items or transactions within the expenditure:
He spent large amounts of money on the house itself, gothicising its
appearance with gargoyles, turrets and panelling.
Bulwer Lytton: The Rise and Fall of a Victorian Man of Letters by Leslie Mitchel, emphasis added
In reference to OP in particular, a singular large amount of sweat would focus on the sweaty body as a whole at a single point of time. The plural expression would add a dimension of complexity: various sweaty places on the body, or perhaps even various causes of sweat over time.
Of course, people do not always hold these subtleties strictly in mind when talking or writing, so it is possible that the plural expression simply adds a sense of emphasis to the extremity of the sweaty condition.