Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?

There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answers your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

https://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

The topic is quite old.

There are three and only three types of rotationally symmetric surfaces for constant $K = -1/a^2$ where $a$ is the cuspidal radius of the central pseudosphere. These are the central pseudosphere (parameter $m =1$) or tractricoid, Conic type ($m < 1$) and Ring type ($m > 1$). The descriptions Ring, Conic type etc. are given by Klein in:

Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). with a reference/reprint iirc from Crelle's Journal.

Asymptotic lines of a Chebyshev Net on these three surfaces are given by the Sine-Gordon Equation.

Present day English translations of Vorlesungen may be available on googling, else one can contact German newsgroups e.g., de.sci.mathematik.

Although the central pseudosphere is often referred to as "the" Beltrami surface, the physical paper model he made is isometrically equivalent to the Conic type ( m > 1) that you can readily verify in Daina's blog. I am discussing it the concurrent thread here

In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him to take his paper models with him when Beltrami returned to University of Pavia 1876.