First off - congratulations on deciding to take control of your finances and begin rebuilding your credit. It's a huge step and you have already made a lot of progress.
About 6.5 years ago, I was in a similar situation. I had 4 SL tradelines and all of them went late for several months. Resulting in about 20 late notations, with several at 120+. I eventually got current on all my loans, and even rehabbed one, but the damage was done, My score was tanked and the lates were like ugly black marks.
Initially I tried writing and calling the SL companies (NelNet and FedLoan) but really got nowhere. The lates were reporting correctly and they said they wouldn't remove them. After a few dozen tries on the phone and a dozen emails, I basically gave up on the goodwill route. I know there have been many others who have had success with asking for removal, but I think its more common to strike out.
Now with regards to your recent score drop, not sure what happened here, but double check your reports to make sure there wasn't something negative added or maybe a huge surge in utilization.
I would avoid the credit repair companies at all costs. They are expensive, and do little more than file disputes with the credit bureaus on your behalf. The gimmick is that if the debt isn't properly validated within the time limit, the negative information will be removed. But I think student loan servicers are likely to be on top of this. So you will likely end up paying $120 bucks to file a dispute that will eventually be thrown out.
So this brings me back to my SL saga, fast forward 6 years and a some of those lates are still there. The final 5-6 lates should be falling off over the next 3 months. My credit score is hovering around 738 these days and I hope to be at 750 once all the lates are gone. Waiting the 7 years was frustrating, but in the grand scheme of things didn't really hold back my rebuild. I did it the same way everyone else did, paying everything on time, not letting more than one balance report, and keeping my utilization low. I can also say that the lates didn't really hold me back from attaining the credit that I wanted and needed. For example, in 2013 my credit was still stuck in the low 630s and I was able to finance a new car with a 2.39% interet rate. Also in 2016, when my credit was 689-690, I bought a 300,000 home using an FHA loan at 3.25% interest. I give these data points to really just show that what seems like the end of the world now, really will start to matter less and less after 12-24 months. I was angry and frustrated when I realized how stupid my mishandling of my credit was. But even without removing my late payments, I still achieved my financial goals by just waiting things out.
So the next 12-24 months might be a little rough, but things will get better slowly over time. Again, best of luck and congratulations on taking control of your credit.
Are these private or federal student loans?
If they're private student loans you might have some luck by sending the bank goodwill letters.
If they're federal student loans you'll have a hard time - servicers are required by law (Higher Education Act, I think) to report complete and accurate information (as opposed to other creditors who are required by the Fair Credit Reporting Act to just report accurate information). There are exceptions to this such as retroactive forbearance, I think.