Mechanism of Fischer esterification: Does the carboxylic acid gives off OH- or H+?
The accepted mechanism is far more likely than your proposed alternate mechanism for a number of reasons.

First, let’s consider an esterification reaction employing a chiral secondary alcohol such as butan-2-ol. Protonation of the alcoholic oxygen is no difficulty and will happen in equilibrium. To generate the corresponding ester from the conjugate acid of butan-2-ol, we would require some nucleophilic attack mechanism: the acid would either attack according to $\mathrm{S_N1}$ or $\mathrm{S_N2}$. If we assume an $\mathrm{S_N2}$-like mechanism, the asymmetric carbon atom would be inverted. If we assume an $\mathrm{S_N1}$ mechanism, it would racemise. Experimental evidence shows, however, that the alcoholic carbon atom strictly retains is stereochemistry — and this principle is used in many total synthesis in which asymmetric alcohols are first generated stereochemically and then transformed into esters of the same stereochemistry. To give you just one of many examples, consider the total synthesis of aetheramide by Gerstmann and Kalesse.[1]


Scheme 1: Example esterification of (R)-butan-2-ol


Scheme 2: Esterification performed during the synthesis of aetheramide by Gerstmann and Kalesse.[1]

Second, and immediately tied to the scheme as referenced above, many alternate esterification methods exist in the literature. In principle, they all work by replacing the acidic particle ($\ce{H+}$) of a Fischer esterification with a less reactive and/or more specific compound. Practically all methods — Steglich or Yamaguchi to name just two — work by adding a compound that is known to interact with the carboxylic acid and not as much with alcohols. However, this is only a weak indication since different methods could follow different mechanisms — even though the quantity of the evidence does seem overwhelming.

Third and finally, the acidic hydrolysis of esters — the reverse reaction of the Fischer esterification — has been studied by $\ce{^18 O}$ marked water. Polanyi and Szabo could show that upon hydrolysis the heavy oxygen atom remains bound to the acid fragment and not to the alcohol.[2] This offers intriguing evidence for the mechanism of nucleophilic attacks on a carbonyl group. I had thought that a corresponding isotope-marking experiment had been performed for the Fischer esterification, too. Unfortunately, I was unable to find a reference. Do note, however, that using isotopically marked oxygen on either the acid or the alcohol would, by careful analysis of the product, rule out or confirm one mechanistic pathway.


Scheme 3: Mechanism of hydrolysis as confirmed by Polanyi and Szabo using $\ce{^18O}$ marked water.[2]



References:

[1]: L. Gerstmann, M. Kalesse, Chem. Eur. J. 2016, 22, 11210–11212. DOI: 10.1002/chem.201602682.

[2]: M. Polanyi, A. L. Szabo, Trans. Faraday Soc. 1934, 30, 508–12. DOI: 10.1039/tf9343000508.