One of the first papers on this topic is probably the one by Polchinski and Strominger (Phys.Rev.Lett. 67 (1991) 1681-1684). However, the first paper that springs to my mind when one talks of effective string theories is the one by Luscher and Weisz. (see also http://arxiv.org/abs/hep-lat/0207003) which I will discuss.
They begin with a classical string in D dimensions (with coordinates Xμ, μ=0,1,…,D−1) stretched between X1=0 and X1=r. Work in a static gauge where one identifies the worldsheet time z0=X0 and z1=X1, , the fluctuations about this string (or collective modes of the string) can be parameterized by a (D−2) dimensional vector h. The idea is to write an action in a derivative expansion with the fluctuations subject to Dirichlet boundary conditions at z1=0,r. The initial term is a Polyakov type action. Evaluating the value of the action for the string will lead to an expression for the quark-antiquark potential as a power series in 1r. The nice tweak to this story by Luscher and Weisz was the use of open-closed duality to constrain these coefficients. Their action takes the form
S=TL (r+μ)+S(2)0+S(2)1+…
where the superscript indicates the derivative order, T is the tension, L is the length of the compact time direction and μ is a constant.
S(2)0=12∫d2z ∂ah⋅∂ah,
S(2)1=b(12∫dz0 (∂1h⋅∂1h) |z1=0+12∫dz0 (∂1h⋅∂1h) |z1=r)
They were able to show using open-closed duality that b=0 -- a non-renormalization theorem. They also discuss the four-derivative terms (odd derivative terms are not present.) where they find two terms. Again, open-closed duality relates the two. The amazing part of this story is that up to four-derivative terms, the coefficients are precisely what one would get by expanding the Nambu-Goto term about the background. In other words, the Nambu-Goto action reproduces the effective quark-antiquark potential much better than one should expect.
Ofer Aharony and his students/post-docs/collaborators have been carrying out a systematic study of extending this work to higher derivatives, non-static gauges and so on. This paper was probably the first in that series. Here they show that there is only one unfixed coefficient (out of three) at sixth-order on imposing (I think) open-closed duality and for D=3 all coefficients get fixed. The introduction to this paper is probably a better summary of things.
I don't think quantizing these effective strings would be a way to get around the c=1 barrier.