The number of trading cards produced during the 1986-93 era was in the millions, so the likelihood that any will appreciate substantially over the next few decades is slim. A much smaller number of insert cards was made. If you can sort those out, it may be worth your time. You should also save any autographed cards.
You might find it more helpful to rely on Ebay completed listings rather than Beckett's as a price guide. Card values provided by Beckett's are not accurate. Many cards sold for less than half their Beckett price even in the early 1990's heyday.
Prepare yourself to accept less than they may be worth if you choose to sell the whole thing. Towards this assumption, any dealer will make an offer taking into account the possibility of receiving thousands of useless "common" cards. It may even be quite challenging for most dealers to make an offer at all as sorting cards takes so much time. The value of time is immeasurable. A large proportion of dealers have a large stock of cards from the junk period. The trash has been thrown in some cases.
You should definitely sift through your old cards if they are from before 1986. Even the "commons" were produced in fewer quantities, which increased their collectability and value.