What make and model is your 4 year old laptop?
Just checking in case it's one of those laptops with 2 or more battery options, with your particular example having had the lowered powered battery from new.
Although I suspect you probably have a gaming laptop, where the gaming performance was created at the expense of battery life. This would make it a fine computer for doing assignments and projects at your home-base.
What will your mobility profile be at uni? As in how much walking & public transport will you be doing with your laptop? And how physically strong are you on the dainty to Olympic weightlifter scale?
This will determine the best size laptop for you.
From there you can look out for deals on specific laptops.
There's no value in buying new laptops. Consumer grade laptops tend to be plasticky and prone to hinges, lids failing from everyday use. As well as the keyboards being quite nasty to type on. Premium business laptops is where you can find the engineering quality, but they're too expensive new. Depreciation on business IT equipment is a wonderful thing for the buyer.
Refurbished is largely a marketing con. Refurbished may mean something on a 1960's Rolls Royce. On a laptop it means nothing of significance. All they do to refurbish laptops is to remove any stickers, run the built-in diagnostics, give it the once over with some furniture cleaner. A refurbished laptop won't be any more reliable nor long lasting than a fully working non-refurbished laptop.