Do there exist enzymes that can take up multiple cofactors to do different reactions?
It sounds like you are asking about what are commonly referred to as "multifunctional enzymes".  

For a reasonably recent article covering this subject — see: 
Cheng, X. Y., Huang, W. J., Hu, S. C., Zhang, H. L., Wang, H., Zhang, J. X., ... &amp; Ji, Z. L. (2012). A global characterization and identification of multifunctional enzymes. PloS one, 7(6), e38979.



A specific example of such an enzyme is the fatty-acyl-CoA synthase from yeast — this enzyme catalyzes multiple reaction steps that use both NADP and Coenzyme A and also uses a third cofactor, FMN. 


  The enzyme from yeasts (Ascomycota and Basidiomycota) is a multi-functional protein complex composed of two subunits. One subunit catalyses the reactions EC 1.1.1.100, 3-oxoacyl-[acyl-carrier-protein] reductase and EC 2.3.1.41, 3-oxoacyl-[acyl-carrier-protein] synthase, while the other subunit catalyses the reactions of EC 2.3.1.38, [acyl-carrier-protein] S-acetyltransferase, EC 2.3.1.39, [acyl-carrier-protein] S-malonyltransferase, EC 4.2.1.59, 3-hydroxypalmitoyl-[acyl-carrier-protein] dehydratase, EC 1.3.1.10, enoyl-[acyl-carrier-protein] reductase (NADPH, Si-specific) and EC 1.1.1.279, (R)-3-hydroxyacid ester dehydrogenase. The enzyme differs from the animal enzyme (EC 2.3.1.85) in that the enoyl reductase domain requires FMN as a cofactor, and the ultimate product is an acyl-CoA (usually palmitoyl-CoA) instead of a free fatty acid.


Source: https://www.ebi.ac.uk/intenz/query?cmd=SearchEC&amp;ec=2.3.1.86