I third Weymouth......but I live here! I’m sorry to hear of your loss.
I would suggest that you book a holiday at one of the caravan parks but as you don’t have a car, the “Waterside Holiday Park & Spa” at Bowleaze Cove is the nearest to the beaches, is rated 5* and lots of things to do - there is a pool, children’s entertainment, beach bar, spa....oh and a Spar supermarket, amusement arcade, funfair rides, cafes all on the doorstep. There is a bus service to the town centre shops and harbour - they might enjoy getting a bucket and go crabbing....the crabs love bacon - but you could even take a long, leisurely walk there along the Esplanade if you fancy it!
For staying at any of the holiday parks It would be best to take a taxi from the train station but Waterside and the Haven caravan park at Preston would only be an 8-10 minute journey from the station. Personally I wouldn’t stay at Chesil Vista - tiny and too far from any action/attractions and Osmington too far if you don’t have a car.
AFAIK you can book ‘The Waterside’ via Hoseasons and www.watersideholidaygroup.co.uk
but some caravan owners let them out privately, which should be cheaper. I think you would need to google for the privately owned ones, I guess some let them out themselves or others use booking agencies. There are also Haven caravan parks but not as well located, especially if you don’t have use of a car. Weymouth also has a number of hotels, B&Bs, holiday cottages - rent via agencies or AirBnB, and two Premier Inns.
You can take the local bus to Portland and the children may enjoy going up the ‘Portland Bill’ lighthouse. There is also a bus to Monkey World at Wareham and the Bovington Tank Museum.
You could spend a day at Lyme Regis or Charmouth by taking the double decker bus which I think is called the ‘Coastal Breezer’ (I did this last Summer on my own, it’s a long journey but there are lovely scenic views). You get the bus from the Esplanade near the King George’s statue.
For bad weather days you can take the train (2 stops) or the no 10 bus to Dorchester as there are a number of museums there (Tutankhamen/Dorset Museum/Shire Hall Court/Teddy Bear museum) and 2 cinemas (there is a Cineworld in Weymouth).