What canI do if I moved 3 months ago and the moving company damaged $5000 worth of furniture but have only offered $250 to cover my loss?

You can sue the moving company for negligence.  An employer is liable for the negligence of its employee(s) which occurred during the course and scope of employment.

Negligence is failure to exercise due care (that degree of care that in this case a reasonable moving company would have exercised under the same or similar circumstances to prevent foreseeable harm). 

In order to prove negligence, you will need to prove duty (of due care mentioned above), breach of duty (failure to exercise due care), actual cause, proximate cause and damages.  Actual cause means but for the movers moving your furniture would your furniture have been damaged?  If the answer is no, which appears to be the case, actual cause has been established.  Proximate cause means were there any unforeseeable, intervening events which would relieve the moving company of liability?  If the answer is no, proximate cause has been established.  Damages means the amount of compensation you are seeking to recover in your lawsuit for negligence against the moving company and movers.  Your damages would be the replacement cost or if applicable cost of repairs to your furniture.  You will need to mitigate (minimize) damages.  For example, replacement cost would have to be comparable to the damaged items.  If you were to select the most expensive items you could find to replace the damaged items, your damages would be reduced accordingly.  If you were to select the most expensive place you could find to repair the damaged items, your damages would be reduced accordingly.  To mitigate damages, a repair shop's charges would have to be comparable to other repair shops in the area.

You will need to file your lawsuit prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter. 

Depending on the total amount of damages you are seeking as compensation, you might be able to file your lawsuit in Small Claims Court.  Your damages should also include the court filing fee and process server fee.

When a mover damages your belongings, they may be liable for the repair or replacement cost (unless, that is, you signed some contract or agreement which limits or puts a cap on their potential liability; check any documents you signed). If they won't pay voluntarily, then you could sue them for the value. For $5,000, if you don't want to or can't afford to foot the cost of an attorney (though note: some attorneys will work on contingency, so they are paid a percentage of whatever they recover), you could sue in small claims court (if the threshhold of your small claims court is less than $5,000, you could sue for up to whatever the threshhold is). The filing fees are low, the court is more informal, and you can act as your own attorney (called being "pro se") to save costs. If you contact the court, either in person or online, you should be able to find more information and sample forms.