If the loans are installment loans or mortgages, paying them down won't increase your score. If they are revolving lines or credit cards, it could help your score by paying them down and you would still have access to the line if you needed it. If they aren't revolving lines of credit or credit cards (only mc, visa, disc, or AE), I would hold on to the money in case you need it. Take the amount that you have saved up and divide it by the number of months you plan on being in school; this will tell you how much you could take from the money you saved to offset the decrease in income.
Since you would be waiting for the 60 day late to fall off, you could start the mod process just to see if you could get that done. Since that process will take some time and you won't be refi'ing, you might as well get in line and see what may happen with your request.