Heavy could also be walnut or mahogany, or anything veneered onto a heavy wood. Pictures of the grain, fairly close up will help identify.

Check the slide hardware for a date or a patent or a company name: it means the table was made AFTER that date (not on that date), or after that company was founded.

Before you break out the sanding equipment try minimally invasive cleaning. You can always decide to sand later, you can't un-sand something.

Mild soap and water ... removes water-soluble dirt, such as sugar and syrup spills. Rub gently and wipe with a clean damp cloth to remove soap, then dry it with another soft cloth.

Mineral spirits and 0000 steel wool ... removes oily dirt, excess wax, hand grime. Rub very gently and wipe off the grunge with a clean soft cloth.

STOP HERE and evaluate your piece. Come back with pics of the cleaned piece and ask what to do next.