Okay, this is my opinion... "Plugging" a leak in an earth dam is really hard. By their design, they leak. But it may be possible (and necessary for safety) to limit the leak and keep it from worsening.
Before you start trying to fix it, I'd suggest some observation and research to make sure you understand what you are trying to fix and what tends to be successful (as a lazy person myself, I only wanna fix things once). It will also give you some reference to know when things are getting worse (if there is a risk of failure), and if you need to call in a professional.
These are some questions that I would be asking myself if I was you:
-How tall is the dam? How much water is it impounding (height, not just surface area)
-What is downstream of it(up to a few miles) What is the consequence if it breaks, and who would be liable?
-What was it built of/how was it built - does it have any sort of core or protective outer layer?
-What maintenance has been done? Is it registered/listed with the state, or have you ever had an inspection done on it? What are the laws in your state/county for dams, dam inspections, maintenance, and what permits are needed for modifications/repairs?
At the dam itself, I would probably go for a walk. I'd look for:
-Is the seepage clear, murky?
-How far up on the dam is it ? Is it at the base ? Halfway up? (1/3 up would be typical).
-Has the flow increased since the last time you were there? What would you estimate the flow to be (garden hose, fire hose, kitchen tap, leaky faucet)? Is there any standing water at the base?
-What is the state of the vegetation on the dam? Does it change? Is there less vegetation at the bottom? Or more, because it is continuously damp?
-Is the dam "straight"? Are there any obvious bulges or depressions?
-What is the soil made of - can you feel it, does it have any sand, gravel? Wet some, roll it in a ball. Does it roll? (clayey) Does it crack (silty) does it roll but shimmer with water? (clay and silt).
-Is the area rocky? Is the bedrock close to surface? (it's possible that the water is coming through the foundation, not your dam itself)
After doing all of this, I would try search terms like "seepage" "earthfill dam" "embankment dam" and "piping failure". Essentially, you don't need a hole on the upstream side to see water coming out in a hole on the downstream side until things get REALLY bad. You will probably also see some stuff about
berms, drains, and blankets being used a solutions. I'd also google "filter" - the idea is to to use a mixture of grain sizes to trap your dam in place by not letting the finer stuff wash away with the water and make a bigger hole.
Hopefully this will give you a better understanding of WHY your dam is leaking, and how people normally fix it... When you are deciding what to do, I'd also keep in mind (whatever you decide to do) that water is persistent - if you block one path, it will very readily move over and start seeping somewhere else. With dams, you want to do things as uniformly as possible, so pressure doesn't concentrate in any one area and cause it to fail.