Let me start by discussing a bit the option of having a large class of generators. You might be interested in the notion of **locally class-presentable**.

To be precise here, I need to be a bit set-theoretical, thus, let me start with an informal comment.

**Informal comment.** Indeed your category is locally class-presentable, class-accessibility is a very strong weakening of the notion of accessibility. On one hand, it escapes the world of categories with a dense (small) generator, on the other, it still allows to **build** on the technical power of the **small object argument** via a **large class of generators**.

**Locally class-presentable and class-accessible categories**, *B. Chorny and J. Rosický*, J. Pure Appl. Alg. 216 (2012), 2113-2125.

discusses a part of the general theory of class-accessibility and class-local presentability. Unfortunately, the paper is designed towards a homotopical treatment and thus insists on weak factorization systems and injectivity but a lot of techniques coming from the classical theory can be recast in this setting.

**Formal comment**. To be mathematically precise, your category is locally large, while locally class-presentable categories will be locally small. Here there are two options, the first one is to study **small sheaves** $\mathsf{Shv}_{\text{small}}(\text{AffSch})$, this is a full subcategory of the category of sheaves and contains many relevant sheaves you want to study. In the informal comment, this is the locally class-presentable one. Two relevant paper to mention on this topic are:

**Exact completions and small sheaves**, *M. Shulman*, Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173.
**Limits of small functors**, *B. J. Day and S. Lack*, Journal of Pure and Applied Algebra, 210(3):651-683, 2007.

The other option is the one of being very careful with universes, in fact restricting to small presheaves might destroy sometimes your only chance of having a right adjoint. It would be too long to elaborate on this last observation here. As a general remark, small presheaves will give you the free completion under small colimits, while all presheaves will give you the free completion under large colimits, how big you need to go depends on the type of constructions that you need to perform.

Coming to the **adjoint functor theorem**, let me state the most general version that I know of. Since it is an *if and only if*, I hope it provides you with a good intuition on when one can expect a right adjoint to exists. The dual version is true for functors preserving limits.

**Thm. (AFT)** Let $f: \mathsf{A} \to \mathsf{B}$ be a functor preserving colimits from a cocomplete category. The following are equivalent:

- For every $b \in \mathsf{B}$, $\mathsf{B}(f\_,b): \mathsf{A}^\circ \to \mathsf{Set}$ is a small presheaf.
- $f$ has a right adjoint.

This version of the AFT is designed for locally small categories and can be made enrichment sensitive - and thus work also for locally large categories - using the proper notion of smallness, or equivalently choosing the correct universe.

Unfortunately, I do not know a reference for this version of the AFT. Indeed it can be deduced by the too general **Thm 3.25** in **On the unicity of formal category theories** by *Loregian and myself*, where it appears as a version of the *very formal adjoint functor theorem* by *Street and Walters* in the language of the preprint.

Finally, about to the **evaluation functor**, I am not an expert of the abelian world, but it looks to me that one can mimic the argument presented in the accepted answer to this question (at least if the topology is subcanonical). Thus, the left adjoint should indeed exist.

https://math.stackexchange.com/questions/2187846/adjoints-to-the-evaluation-functors

This answer is closely connected to this other.